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Block library - User contributions [en-gb]
2026-05-08T12:04:37Z
User contributions
MediaWiki 1.30.1
http://wiki.manchester.ac.uk/blocks/index.php?title=Q16&diff=1248
Q16
2025-04-02T15:05:50Z
<p>Charles Eaton: Removed incorrect statement of Holm result.</p>
<hr />
<div>__NOTITLE__<br />
<br />
== Blocks with defect group <math>Q_{16}</math> ==<br />
<br />
These are examples of [[Tame blocks|tame blocks]] and were first classified over <math>k</math> by Erdmann (see [[References|[Er88a], [Er88b]]]) with some exceptions.<ref>Note that as in [[References|[Ho97]]] the class <math>Q(2 {\cal B})_2</math> cannot be realised by a block.</ref> It is not known which algebras in the infinite families <math>Q(2 {\cal A})</math> and <math>Q(2 {\cal B})_1</math> are realised by blocks, and as such Donovan's conjecture is still open for <math>Q_{16}</math> for blocks with two simple modules. Until this is resolved the labelling is provisional.<br />
<br />
For blocks with three simple modules the <math>k</math>-Morita equivalence classes lift to unique <math>\mathcal{O}</math>-classes by [[References|[Ei16]]], but otherwise the classification with respect to <math>\mathcal{O}</math> is still unknown.<br />
<br />
'''<pre style="color: red">CLASSIFICATION INCOMPLETE</pre>'''<br />
<br />
{| class="wikitable"<br />
|-<br />
! scope="col"| Class<br />
! scope="col"| Representative<br />
! scope="col"| # lifts / <math>\mathcal{O}</math><br />
! scope="col"| <math>k(B)</math><br />
! scope="col"| <math>l(B)</math><br />
! scope="col"| Inertial quotients<br />
! scope="col"| <math>{\rm Pic}_\mathcal{O}(B)</math><br />
! scope="col"| <math>{\rm Pic}_k(B)</math><br />
! scope="col"| <math>{\rm mf_\mathcal{O}(B)}</math><br />
! scope="col"| <math>{\rm mf_k(B)}</math><br />
! scope="col"| Notes<br />
<br />
|-<br />
|[[M(16,9,1)]] || <math>kSD_{16}</math> || 1 ||7 ||1 ||<math>1</math> || || || ||1 ||<br />
|-<br />
|[[M(16,9,2)]] || <math>B_0(k \tilde{S}_5)</math><ref>This is the double cover SmallGroup(240,89)</ref> || ? ||8 ||2 ||<math>1</math> || || || ||1 || <math>Q(2 {\cal A})</math><br />
|-<br />
|[[M(16,9,3)]] || <math>B_0(k \tilde{S}_4)</math><ref>This is the double cover SmallGroup(48,28)</ref> || ? ||8 ||2 ||<math>1</math> || || || ||1 || <math>Q(2 {\cal B})_1</math><br />
|-<br />
|[[M(16,9,4)]] || <math>B_0(kSL_2(9))</math> || 1 ||9 ||3 ||<math>1</math> || || || ||1 || <math>Q(3 {\cal A})_2</math><br />
|-<br />
|[[M(16,9,5)]] || <math>B_0(k(2.A_7))</math> || 1 ||10 ||3 ||<math>1</math> || || || ||1 || <math>Q(3 {\cal B})</math><br />
|-<br />
|[[M(16,9,6)]] || <math>B_0(kSL_2(7))</math> || 1 ||9 ||3 ||<math>1</math> || || || ||1 || <math>Q(3 {\cal K})</math><br />
|}<br />
<br />
[[M(16,9,2)]] and [[M(16,9,3)]] are derived equivalent over <math>k</math> by [[References|[Ho97]]].<br />
<br />
[[M(16,9,4)]], [[M(16,9,5)]] and [[M(16,9,6)]] are derived equivalent over <math>\mathcal{O}</math> by [[References|[Ei16]]]<ref>This result was obtained over <math>k</math> in [[References|[Ho97]]]</ref>.<br />
<br />
== Notes ==<br />
<br />
<references /></div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Q16&diff=1247
Q16
2025-04-02T15:02:39Z
<p>Charles Eaton: Q(2B)_1</p>
<hr />
<div>__NOTITLE__<br />
<br />
== Blocks with defect group <math>Q_{16}</math> ==<br />
<br />
These are examples of [[Tame blocks|tame blocks]] and were first classified over <math>k</math> by Erdmann (see [[References|[Er88a], [Er88b]]]) with some exceptions.<ref>Note that as in [[References|[Ho97]]] the class <math>Q(2 {\cal B})_2</math> cannot be realised by a block.</ref> It is not known which algebras in the infinite families <math>Q(2 {\cal A})</math> and <math>Q(2 {\cal B})_1</math> are realised by blocks, and as such Donovan's conjecture is still open for <math>Q_{16}</math> for blocks with two simple modules. Until this is resolved the labelling is provisional.<br />
<br />
For blocks with three simple modules the <math>k</math>-Morita equivalence classes lift to unique <math>\mathcal{O}</math>-classes by [[References|[Ei16]]], but otherwise the classification with respect to <math>\mathcal{O}</math> is still unknown.<br />
<br />
'''<pre style="color: red">CLASSIFICATION INCOMPLETE</pre>'''<br />
<br />
{| class="wikitable"<br />
|-<br />
! scope="col"| Class<br />
! scope="col"| Representative<br />
! scope="col"| # lifts / <math>\mathcal{O}</math><br />
! scope="col"| <math>k(B)</math><br />
! scope="col"| <math>l(B)</math><br />
! scope="col"| Inertial quotients<br />
! scope="col"| <math>{\rm Pic}_\mathcal{O}(B)</math><br />
! scope="col"| <math>{\rm Pic}_k(B)</math><br />
! scope="col"| <math>{\rm mf_\mathcal{O}(B)}</math><br />
! scope="col"| <math>{\rm mf_k(B)}</math><br />
! scope="col"| Notes<br />
<br />
|-<br />
|[[M(16,9,1)]] || <math>kSD_{16}</math> || 1 ||7 ||1 ||<math>1</math> || || || ||1 ||<br />
|-<br />
|[[M(16,9,2)]] || <math>B_0(k \tilde{S}_5)</math><ref>This is the double cover SmallGroup(240,89)</ref> || ? ||8 ||2 ||<math>1</math> || || || ||1 || <math>Q(2 {\cal A})</math><br />
|-<br />
|[[M(16,9,3)]] || <math>B_0(k \tilde{S}_4)</math><ref>This is the double cover SmallGroup(48,28)</ref> || ? ||8 ||2 ||<math>1</math> || || || ||1 || <math>Q(2 {\cal B})_1</math><br />
|-<br />
|[[M(16,9,4)]] || <math>B_0(kSL_2(9))</math> || 1 ||9 ||3 ||<math>1</math> || || || ||1 || <math>Q(3 {\cal A})_2</math><br />
|-<br />
|[[M(16,9,5)]] || <math>B_0(k(2.A_7))</math> || 1 ||10 ||3 ||<math>1</math> || || || ||1 || <math>Q(3 {\cal B})</math><br />
|-<br />
|[[M(16,9,6)]] || <math>B_0(kSL_2(7))</math> || 1 ||9 ||3 ||<math>1</math> || || || ||1 || <math>Q(3 {\cal K})</math><br />
|}<br />
<br />
[[M(16,9,2)]] and [[M(16,9,3)]] are derived equivalent over <math>k</math> by [[References|[Ho97]]], in which it is further proved that ''all'' blocks with defect group <math>Q_{16}</math> and two simple modules are derived equivalent (irrespective of the unknown cases in the classification).<br />
<br />
[[M(16,9,4)]], [[M(16,9,5)]] and [[M(16,9,6)]] are derived equivalent over <math>\mathcal{O}</math> by [[References|[Ei16]]]<ref>This result was obtained over <math>k</math> in [[References|[Ho97]]]</ref>.<br />
<br />
== Notes ==<br />
<br />
<references /></div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Fusion-trivial_p-groups&diff=1246
Fusion-trivial p-groups
2024-05-02T15:50:10Z
<p>Charles Eaton: </p>
<hr />
<div>A p-group <math>P</math> is ''p-nilpotent forcing'' if any finite group <math>G</math> that contains <math>P</math> as a Sylow p-subgroup must be p-nilpotent (that is <math>G=O_{p'}(G)P</math>). These groups appear in [[References#W|[vdW91]]].<br />
<br />
There does not seem to be any name given to p-groups <math>P</math> for which the only saturated fusion system is <math>\mathcal{F}_P(P)</math>. We will refer to them as ''fusion-trivial p-groups'' (although appropriate names might also be ''nilpotent forcing'' or ''fusion nilpotent forcing''). Examples of such p-groups are abelian 2-groups <math>P</math> for which <math>{\rm Aut}(P)</math> is a 2-group, i.e., those abelian 2-groups whose cyclic direct factors have pairwise distinct orders.<br />
<br />
A p-group <math>P</math> is fusion-trivial if and only if it is [[Glossary|resistant]] and <math>{\rm Aut}(P)</math> is a p-group. Recall that <math>P</math> is resistant if whenever <math>\mathcal{F}</math> is a saturated fusion system on <math>P</math>, we have <math>\mathcal{F}=N_{\mathcal{F}}(P)</math>, or equivalently <math>\mathcal{F}=\mathcal{F}_P(G)</math> for some finite group <math>G</math> with <math>P</math> as a normal Sylow p-subgroup. Resistant p-groups were introduced in [[References#S|[St02]]] in terms of fusion systems for groups, and for arbitrary saturated fusion systems in [[References#S|[St06]]].<br />
<br />
Theorem 4.8 of [[References#S|[St06]]] yields a useful necessary and sufficient condition for a p-group <math>P</math> to be resistant: there exists a central series \[P=P_n \geq P_{n-1} \geq \cdots \geq P_1\] for which each <math>P_i</math> is weakly <math>\mathcal{F}</math>-closed in any saturated fusion system on <math>P</math>. This happens for example if each <math>P_i</math> is the unique subgroup of <math>P</math> of its isomorphism type.<br />
<br />
Following [[References#M|[Ma86]]] (and [[References#H|[HM07]]]), Henn and Priddy proved in [[References#H|[HP94]]] that in some sense asymptotically most <math>p</math>-groups only occur as Sylow <math>p</math>-subgroups of <math>p</math>-nilpotent groups. In [[References#T|[Th93]]] it was proved that the <math>p</math>-groups considered in [[References#H|[HP94]]] have a strongly characteristic central series, in which each term is the unique subgroup of its isomorphism type. Hence in the sense of [[References#M|[Ma86]]], almost every <math>p</math>-group is fusion trivial. This leaves the natural question of whether a version of this result with a cleaner definition of "almost all" holds:<br />
<br />
<div class="boxed"><br />
=== Question on fusion-trivial p-groups ===<br />
Does the proportion of p-groups of order <math>p^n</math> that are fusion-trivial tend to 1 as <math>n \rightarrow \infty</math>? <br />
</div></div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Fusion-trivial_p-groups&diff=1245
Fusion-trivial p-groups
2024-05-02T15:48:23Z
<p>Charles Eaton: </p>
<hr />
<div>A p-group <math>P</math> is ''p-nilpotent forcing'' if any finite group <math>G</math> that contains <math>P</math> as a Sylow p-subgroup must be p-nilpotent (that is <math>G=O_{p'}(G)P</math>). These groups appear in [[References#W|[vdW91]]].<br />
<br />
There does not seem to be any name given to p-groups <math>P</math> for which the only saturated fusion system is <math>\mathcal{F}_P(P)</math>. We will refer to them as ''fusion-trivial p-groups'' (although appropriate names might also be ''nilpotent forcing'' or ''fusion nilpotent forcing''). Examples of such p-groups are abelian 2-groups <math>P</math> for which <math>{\rm Aut}(P)</math> is a 2-group, i.e., those abelian 2-groups whose cyclic direct factors have pairwise distinct orders.<br />
<br />
A p-group <math>P</math> is fusion-trivial if and only if it is [[Glossary|resistant]] and <math>{\rm Aut}(P)</math> is a p-group. Recall that <math>P</math> is resistant if whenever <math>\mathcal{F}</math> is a saturated fusion system on <math>P</math>, we have <math>\mathcal{F}=N_{\mathcal{F}}(P)</math>, or equivalently <math>\mathcal{F}=\mathcal{F}_P(G)</math> for some finite group <math>G</math> with <math>P</math> as a normal Sylow p-subgroup. Resistant p-groups were introduced in [[References#S|[St02]]] in terms of fusion systems for groups, and for arbitrary saturated fusion systems in [[References#S|[St06]]].<br />
<br />
Theorem 4.8 of [[References#S|[St06]]] yields a useful necessary and sufficient condition for a p-group <math>P</math> to be resistant: there exists a central series \[P=P_n \geq P_{n-1} \geq \cdots \geq P_1\] for which each <math>P_i</math> is weakly <math>\mathcal{F}</math>-closed in any saturated fusion system on <math>P</math>. This happens for example if each <math>P_i</math> is the unique subgroup of <math>P</math> of its isomorphism type.<br />
<br />
Following [[References#M|[Ma86]]] (and [[References#H|[HM07]]]), Henn and Priddy proved in [[References#H|[HP94]]] that in some sense asymptotically most <math>p</math>-groups only occur as Sylow <math>p</math>-subgroups of <math>p</math>-nilpotent groups. In [[References#T|[Th93]]] it was proved that the <math>p</math>-groups considered in [[References#H|[HP94]]] have a strongly characteristic central series, in which each term is the unique subgroup of its isomorphism type. Hence in the sense of [[References#M|[Ma86]]], almost every <math>p</math>-group is fusion trivial. This leaves the natural question of whether a version of this result with a cleaner definition of "almost all" holds:<br />
<br />
<div class="boxed"><br />
=== Question on fusion-trivial p-groups ===<br />
Does the proportion of p-groups of order <math>p^n</math> that are fusion-trivial tend to 1 as <math>n \rightarrow \infty</math>? <br />
</div><br />
<br />
In practice, a more realistic question would mimic the asymptotic results mentioned above.</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Main_Page&diff=1244
Main Page
2024-05-02T12:49:27Z
<p>Charles Eaton: Removed links from banner</p>
<hr />
<div>__NOTOC__<br />
<!-- BANNER ACROSS TOP OF PAGE --><br />
<div id="mp-topbanner" style="clear:both; position:relative; box-sizing:border-box; width:100%; margin:1.2em 0 6px; min-width:40em; border:1px solid #f5b041; background-color:#fad7a0; color:#000; white-space:nowrap;"><br />
<!-- "WELCOME BANNER" --><br />
<div style="margin:0.4em; width:22em; text-align:center;"><br />
<div style="font-size:170%; font-weight:bold; padding:.1em;">Blocks of finite groups</div><br />
<div style="font-size:95%;">Donovan's conjecture and the classification </div><br />
<div style="font-size:95%;">of Morita equivalence classes</div><br />
</div><br />
<!--<ul style="position:absolute; right:-1em; top:50%; margin-top:-2.4em; width:38%; font-size:95%;"><br />
<li style="position:absolute; left:0; top:0;">[[Status of Donovan's conjecture]]</li><br />
<li style="position:absolute; left:0; top:1.6em;">[[Classification by p-group]]</li><br />
<li style="position:absolute; left:0; top:3.2em;">[[Generic classifications by p-group class]]</li><br />
</ul>--><br />
</div><br />
<br />
A resource for progress on [[Statements_of_conjectures#Donovan's conjecture|Donovan's conjecture]] and for the classification of Morita equivalence classes of blocks with a given defect group. The intention is to eventually make the site editable by all, but at the moment it is in its early stages. If you would like to contribute to this site, then please contact [[User:Charles Eaton|Charles Eaton]].<br />
<br />
A more detailed account of the purpose and content of the site can be found [[About the site|here]]. <br />
<br />
{| style="width: 100%; margin-top:4px; border-spacing: 0px;"<br />
| style="width:50%; border:1px solid #a3e4d7; padding:0; background:#f5fffa; vertical-align:top; color:#000;" |<br />
<h2 style="margin:0.5em; background:#cef2e0; font-family:inherit; font-size:120%; font-weight:bold; border:1px solid #a3bfb1; color:#000; padding:0.2em 0.4em;">Classifications</h2><br />
*[[Classification by p-group]]<br />
*[[Generic classifications by p-group class]]<br />
*[[Results on classes of groups]]<br />
*[[Miscallaneous results]]<br />
*[[Fusion-trivial p-groups]]<br />
*[[Open problems]]<br />
<div style="padding:0.1em 0.6em;"></div><br />
<br />
| style="border:1px solid transparent;" |<br />
| style="width:50%; border:1px solid #aed6f1; padding:0; background:#f5faff; vertical-align:top;"|<br />
<h2 style="margin:0.5em; background:#cedff2; font-family:inherit; font-size:120%; font-weight:bold; border:1px solid #a3b0bf; color:#000; padding:0.2em 0.4em;">Conjectures</h2><br />
*[[Statements of conjectures]]<br />
*[[Status of Donovan's conjecture]]<br />
*[[Reductions]]<br />
<div style="padding:0.1em 0.6em;"></div><br />
|}<br />
<br />
{| style="width: 100%; margin-top:4px; border-spacing: 0px;"<br />
| style="width:50%; border:1px solid #d6b8b8; ; padding:0; background:#fff9f9; vertical-align:top; color:#000;" |<br />
<h2 id="mp-tfa-h2" style="margin:0.5em; background:#f9e8e8; font-family:inherit; font-size:120%; font-weight:bold; border:1px solid #b59898; color:#000; padding:0.2em 0.4em;">Resources</h2><br />
*[[Background and conjectures]]<br />
*[[Notation]]<br />
*[[Labelling for Morita equivalence classes]]<br />
*[[Glossary]]<br />
*[[References]]<br />
*[[Other resources|Links to other resources]]<br />
*[[Guide to Contributing]]<br />
*[[To do list]]<br />
<div style="padding:0.1em 0.6em;"></div><br />
<br />
| style="border:1px solid transparent;" |<br />
| style="width:50%; border:1px solid #d7bde2; padding:0; background:#f5eef8; vertical-align:top;"|<br />
<h2 style="margin:0.5em; background: #ebdef0; font-family:inherit; font-size:120%; font-weight:bold; border:1px solid #c39bd3; color:#000; padding:0.2em 0.4em;">Picard groups</h2><br />
*[[Picard groups]]<br />
*[[Known Picard groups]]<br />
*[[Groups of perfect self-isometries]]<br />
<div style="padding:0.1em 0.6em;"></div><br />
|}<br />
<br />
This wiki site was set up by [[User:Charles Eaton|Charles Eaton]] and [[User:CesareGArdito|Cesare G. Ardito]] and is maintained by [[User:Charles Eaton|Charles Eaton]] and hosted at the University of Manchester.</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Morita_invariants&diff=1243
Morita invariants
2024-05-02T11:40:11Z
<p>Charles Eaton: </p>
<hr />
<div>[[Image:under-construction.png|50px|left]]<br />
<br />
== Invariants preserved under Morita equivalence of blocks of f.d. <math>k</math>-algebras ==<br />
<br />
* Number of isomorphism classes of simple modules<br />
* Isomorphism type of centre of algebra, and so dimension of centre (hence, in the case of <math>k</math>-blocks of finite groups, the number of irreducible characters associated to the corresponding <math>\mathcal{O}</math>-block)<br />
* Loewy structure and socle structure<br />
<br />
== Invariants preserved under <math>k</math>-Morita equivalence of blocks of finite groups ==<br />
<br />
* Invariants listed above<br />
* Cartan matrix, up to rearrangement of rows and columns<br />
* Order of defect group<br />
* Exponent of defect group<ref>See (78) of [[References#K|[Ku91]]]</ref><br />
* <math>p</math>-rank of defect group<ref>See Corollary 4 of [[References#A|[AE81]]], attributed to Donovan</ref><br />
<br />
== Invariants preserved under <math>\mathcal{O}</math>-Morita equivalence of blocks of finite groups ==<br />
<br />
* Invariants listed above<br />
* Number <math>k_h(B)</math> of irreducible characters of a given [[Glossary#Height of an irreducible character|height]] <math>h</math><br />
* Decomposition matrix, up to rearrangement of rows and columns<br />
<br />
== Invariants preserved under basic Morita equivalence of blocks of finite groups<ref>Note that by [[References#K|[KL18]]], if <math>k</math>-blocks are basic Morita equivalent, then the corresponding <math>\mathcal{O}</math>-blocks are Morita equivalent</ref> ==<br />
<br />
* Invariants listed above<br />
* Isomorphism type of a defect group<ref>See Corollary 3.6 of [[References#P|[Pu99]]]</ref><br />
* [[Glossary#Fusion system|Fusion system]] of the block<ref>See Corollary 3.6 of [[References#P|[Pu99]]]</ref><br />
<br />
== Invariants preserved under splendid Morita equivalence of blocks of finite groups ==<br />
<br />
* Invariants listed above<br />
* [[Glossary#Source algebra|Source algebra]]<br />
<br />
== Notes ==<br />
<br />
<references /></div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=References&diff=1242
References
2024-05-02T11:29:03Z
<p>Charles Eaton: </p>
<hr />
<div>[[#A|A,]] [[#B|B,]] [[#C|C,]] [[#D|D,]] [[#E|E,]] [[#F|F,]] [[#G|G,]] [[#H|H,]] [[#I|I,]] [[#J|J,]] [[#K|K,]] [[#L|L,]] [[#M|M,]] [[#N|N,]] [[#O|O,]] [[#P|P,]] [[#Q|Q,]] [[#R|R,]] [[#S|S,]] [[#T|T]] [[#U|U,]] [[#V|V,]] [[#W|W,]] [[#X|X,]] [[#Y|Y,]] [[#Z|Z,]] <br />
<br />
{| <br />
|- id="A"<br />
|[Al79] || '''J. L. Alperin''', ''Projective modules for <math>SL(2,2^n)</math>'', J. Pure and Applied Algebra '''15''' (1979), 219-234.<br />
|-<br />
|[Al80] || '''J. L. Alperin''', ''Local representation theory'', The Santa Cruz Conference on Finite Groups., Proc. Sympos. Pure Math. '''37''' (1980), 369-375.<br />
|-<br />
|[AE81] || '''J. L. Alperin and L. Evens''', ''Representations, resoluutions and Quillen's dimension theorem'', J. Pure Appl. Algebra '''22''' (1981), 1-9.<br />
|-<br />
|[AE04] || '''Jianbei An and C. W. Eaton''', ''Blocks with trivial intersection defect groups'', Math. Z. '''247''' (2004), 461-486.<br />
|-<br />
|[AE04] || '''Jianbei An and C. W. Eaton''', [https://arxiv.org/abs/2310.02150 ''Morita equivalence classes of blocks with extraspecial defect groups <math>p_+^{1+2}</math>''], [https://arxiv.org/abs/2310.02150 arxiv:2310.02150]<br />
|-<br />
|[Ar19] || '''C. G. Ardito''', [https://arxiv.org/abs/1908.02652 ''Morita equivalence classes of blocks with elementary abelian defect groups of order 32''], J. Algebra '''573''' (2021), 297-335.<br />
|-<br />
|[ArMcK20] || '''C. G. Ardito and E. McKernon''', ''[https://arxiv.org/abs/2010.08329 ''2-blocks with an abelian defect group and a freely acting cyclic inertial quotient''], [https://arxiv.org/abs/2010.08329 arxiv.org/abs/2010.08329]<br />
|-<br />
|[AS20] || '''C. G. Ardito and B. Sambale''', [http://www.advgrouptheory.com/journal/Volumes/12/ArditoSambale.pdf ''Broué's Conjecture for 2-blocks with elementary abelian defect groups of order 32''], Advances in Group Theory and Applications 12 (2021), 71–78. <br />
|-<br />
|[AKO11] || '''M. Aschbacher, R. Kessar and B. Oliver''', ''Fusion systems in algebra and topology'', London Mathematical Society Lecture Notes '''391''', Cambridge University Press (2011).<br />
|- id="B"<br />
|[BK07] || '''D. Benson and R. Kessar''', ''Blocks inequivalent to their Frobenius twists'', J. Algebra '''315''' (2007), 588-599.<br />
|-<br />
|[BS23] || '''D. Benson and B. Sambale''', [https://arxiv.org/abs/2301.10537 ''Finite dimensional algebras not arising as blocks in group algebras''], [https://arxiv.org/pdf/2301.10537 arxiv:2301.10537]<br />
|-<br />
|[BKL18] || '''R. Boltje, R. Kessar, and M. Linckelmann''', [https://doi.org/10.1016/j.jalgebra.2019.02.045 ''On Picard groups of blocks of finite groups''], J. Algebra '''558''' (2020), 70-101.<br />
|-<br />
|[Bra41] || '''R. Brauer''', ''Investigations on group characters'', Ann. Math. '''42''' (1941), 936-958.<br />
|-<br />
|[BP80] || '''M. Broué and L. Puig''', ''A Frobenius theorem for blocks'', Invent. Math. '''56''' (1980), 117-128.<br />
|-<br />
|[BP80b] || '''M. Broué and L. Puig''', ''Characters and local structure in G-algebras'', J. Algebra '''63''' (1980), 306-317.<br />
|- id="C"<br />
|[Cr11] || '''D. A. Craven''', ''The Theory of Fusion Systems: An Algebraic Approach'', Cambridge University Press (2011).<br />
|-<br />
|[Cr12] || '''D. A. Craven''', [https://arxiv.org/abs/1207.0116 ''Perverse Equivalences and Broué's Conjecture II: The Cyclic Case''], [https://arxiv.org/abs/1207.0116 arXiv:1207.0116]<br />
|-<br />
|[CDR18] || '''D. A. Craven, O. Dudas and R. Rouquier''', [https://arxiv.org/abs/1701.07097 ''The Brauer trees of unipotent blocks''], to appear, J. EMS, [https://arxiv.org/abs/1701.07097 arXiv:1701.07097] <br />
|-<br />
|[CEKL11] || '''D. A. Craven, C. W. Eaton, R. Kessar and M. Linckelmann''', ''The structure of blocks with a Klein four defect group'', Math. Z. '''268''' (2011), 441-476.<br />
|-<br />
|[CG12] || '''D. A. Craven and A. Glesser''', ''Fusion systems on small p-groups'', Trans. AMS '''364''' (2012) 5945-5967.<br />
|-<br />
|[CR13] || '''D. A. Craven and R. Rouquier''', ''Perverse equivalences and Broué's conjecture'', Adv. Math. '''248''' (2013), 1-58.<br />
|-<br />
|[CuRe81a] || '''C. W. Curtis and I. Reiner''', ''Methods of representation theory, with applications to finite groups and orders, Volume I'', Wiley-Interscience (1981).<br />
|-<br />
|[CuRe81b] || '''C. W. Curtis and I. Reiner''', ''Methods of representation theory, with applications to finite groups and orders, Volume II'', Wiley-Interscience (1981).<br />
|- id="D"<br />
|[Da66] || '''E. C. Dade''', ''Blocks with cyclic defect groups'', Ann. Math. '''84''' (1966), 20-48. <br />
|-<br />
|[DE20] || '''S. Danz and K. Erdmann''', [https://arxiv.org/abs/2008.10999 ''On Ext-Quivers of Blocks of weight two for symmetric groups''], [https://arxiv.org/abs/2008.10999 arXiv:2008.10999]<br />
|-<br />
|[Du14] || '''O. Dudas''', [https://arxiv.org/abs/1011.5478 ''Coxeter orbits and Brauer trees II''], Int. Math. Res. Not. '''15''' (2014), 4100-4123.<br />
|-<br />
|[Dü04] || '''O. Düvel''', ''On Donovan's conjecture'', J. Algebra '''272''' (2004), 1-26.<br />
|- id="E"<br />
|[Ea16] || '''C. W. Eaton''', ''Morita equivalence classes of <math>2</math>-blocks of defect three'', Proc. AMS '''144''' (2016), 1961-1970.<br />
|-<br />
|[Ea18] || '''C. W. Eaton''', [https://arxiv.org/abs/1612.03485 ''Morita equivalence classes of blocks with elementary abelian defect groups of order 16''], [https://arxiv.org/abs/1612.03485 arXiv:1612.03485]<br />
|-<br />
|[Ea24] || '''C. W. Eaton''', [https://arxiv.org/abs/2401.04028 ''Blocks whose defect groups are Suzuki 2-groups''], [https://arxiv.org/abs/2401.04028 arXiv:2401.04028]<br />
|-<br />
|[EEL18] || '''C. W. Eaton, F. Eisele and M. Livesey''', [https://arxiv.org/abs/1809.08152 ''Donovan's conjecture, blocks with abelian defect groups and discrete valuation rings''], Math. Z. '''295''' (2020), 249-264.<br />
|-<br />
|[EKKS14] || '''C. W. Eaton, R. Kessar, B. Külshammer and B. Sambale''', ''<math>2</math>-blocks with abelian defect groups'', Adv. Math. '''254''' (2014), 706-735.<br />
|-<br />
|[EKS12] || '''C. W. Eaton, B. Külshammer and B. Sambale''', ''<math>2</math>-blocks with minimal nonabelian defect groups, II'', J. Group Theory '''15''' (2012), 311-321.<br />
|-<br />
|[EL18a] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/1709.04331 Classifying blocks with abelian defect groups of rank 3 for the prime 2]'', J. Algebra '''515''' (2018), 1-18.<br />
|-<br />
|[EL18b] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/1803.03539 Donovan's conjecture and blocks with abelian defect groups]'', Proc. AMS. '''147''' (2019), 963-970.<br />
|-<br />
|[EL18c] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/1810.10950 Some examples of Picard groups of blocks]'', J. Algebra '''558''' (2020), 350-370.<br />
|-<br />
|[EL20] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/2006.11173 Donovan's conjecture and extensions by the centralizer of a defect group]'', J. Algebra '''582''' (2021), 157-176.<br />
|-<br />
|[EL23] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/2310.05734 Morita equivalence classes of <math>2</math>-blocks with abelian defect groups of rank <math>4</math>]'', [https://arxiv.org/abs/2310.05734 arxiv:2310.05734], to appear, J. LMS<br />
|-<br />
|[Ei16] || '''F. Eisele''', ''Blocks with a generalized quaternion defect group and three simple modules over a <math>2</math>-adic ring'', J. Algebra '''456''' (2016), 294-322.<br />
|-<br />
|[Ei18] || '''F. Eisele''', ''[https://arxiv.org/abs/1807.05110 The Picard group of an order and Külshammer reduction]'', Algebr. Represent. Theory '''24''' (2021), 505-518.<br />
|-<br />
|[Ei19] || '''F. Eisele''', ''[https://arxiv.org/abs/1908.00129 On the geometry of lattices and finiteness of Picard groups]'', J. Reine Angew. Math. '''782''' (2022), 219-333. <br />
|-<br />
|[EiLiv20] || '''F. Eisele and M. Livesey''', ''[https://arxiv.org/abs/2006.13837 Arbitrarily large Morita Frobenius numbers]'', Algebra Number Theory '''16''' (2022), 1889-1904.<br />
|-<br />
|[Er82] || '''K. Erdmann''', ''Blocks whose defect groups are Klein four groups: a correction'', J. Algebra '''76''' (1982), 505-518.<br />
|-<br />
|[Er87] || '''K. Erdmann''', ''Algebras and dihedral defect groups'', Proc. LMS '''54''' (1987), 88-114.<br />
|-<br />
|[Er88a] || '''K. Erdmann''', ''Algebras and quaternion defect groups, I'', Math. Ann. '''281''' (1988), 545-560.<br />
|-<br />
|[Er88b] || '''K. Erdmann''', ''Algebras and quaternion defect groups, II'', Math. Ann. '''281''' (1988), 561-582. <br />
|-<br />
|[Er88c] || '''K. Erdmann''', ''Algebras and semidihedral defect groups I'', Proc. LMS '''57''' (1988), 109-150. <br />
|-<br />
|[Er90] || ''' K. Erdmann''', ''Blocks of tame representation type and related algebras'', Lecture Notes in Mathematics '''1428''', Springer-Verlag (1990).<br />
|-<br />
|[Er90b] || '''K. Erdmann''', ''Algebras and semidihedral defect groups II'', Proc. LMS '''60''' (1990), 123-165.<br />
|- id="F"<br />
|[Fa17] || '''N. Farrell''', ''On the Morita Frobenius numbers of blocks of finite reductive groups'', J. Algebra '''471''' (2017), 299-318.<br />
|-<br />
|[FK18] || '''N. Farrell and R. Kessar''', [https://arxiv.org/abs/1805.02015 ''Rationality of blocks of quasi-simple finite groups''], Represent. Theory '''23''' (2019), 325-349.<br />
|- id="G"<br />
|[GMdelR21] || '''D. Garcia, l. Margolis and A. del Rio''', [https://arxiv.org/abs/2016.07231 ''Non-isomorphic 2-groups with isomorphic modular group algebras''], J. Reine Angew. Math. '''f783''' (2022), 269–274.<br />
|-<br />
|[GO97] || '''H. Gollan and T. Okuyama''', ''Derived equivalences for the smallest Janko group'', preprint (1997).<br />
|-<br />
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|- id="H"<br />
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|-<br />
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|-<br />
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|- <br />
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|-<br />
|[HK05] || '''G. Hiss and R. Kessar''', ''Scopes reduction and Morita equivalence classes of blocks in finite classical groups II'', J. Algebra '''283''' (2005), 522-563.<br />
|-<br />
|[Ho97] || '''T. Holm''', ''Derived equivalent tame blocks'', J. Algebra '''194''' (1997), 178-200.<br />
|-<br />
|[HKL07] || '''T. Holm, R. Kessar and M. Linckelmann''', ''Blocks with a quaternion defect group over a 2-adic ring: the case <math>\tilde{A}_4</math>'', Glasgow Math. J. '''49''' (2007), 29–43.<br />
|- id="J"<br />
|[Ja69] || '''G. Janusz''', ''Indecomposable modules for finite groups'', Ann. Math. '''89''' (1969), 209-241.<br />
|-<br />
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|- id="K"<br />
|[Ke96] || '''R. Kessar''', ''Blocks and source algebras for the double covers of the symmetric and alternating groups'', J. Algebra '''186''' (1996), 872-933.<br />
|-<br />
|[Ke00] || '''R. Kessar''', ''Equivalences for blocks of the Weyl groups'', Proc. Amer. Math. Soc. '''128''' (2000), 337-346.<br />
|-<br />
|[Ke01] || '''R. Kessar''', ''Source algebra equivalences for blocks of finite general linear groups over a fixed field'', Manuscripta Math. '''104''' (2001), 145-162. <br />
|-<br />
|[Ke02] || '''R. Kessar''', ''Scopes reduction for blocks of finite alternating groups'', Quart. J. Math. '''53''' (2002), 443-454.<br />
|-<br />
|[Ke05] || ''' R. Kessar''', ''A remark on Donovan's conjecture'', Arch. Math (Basel) '''82''' (2005), 391-394.<br />
|-<br />
|[KL18] || '''R. Kessar and M. Linckelmann''', [https://arxiv.org/abs/1705.07227 ''Descent of equivalences and character bijections''], [https://arxiv.org/abs/1705.07227 arXiv:1705.07227]<br />
|-<br />
|[Ki84] || '''M. Kiyota''', ''On 3-blocks with an elementary abelian defect group of order 9'', J. Fac. Sci. Univ. Tokyo Sect. IA Math. '''31''' (1984), 33–58.<br />
|-<br />
|[Ko03] || '''S. Koshitani''', ''Conjectures of Donovan and Puig for principal <math>3</math>-blocks with abelian defect groups'', Comm. Alg. '''31''' (2003), 2229-2243; ''Corrigendum'', '''32''' (2004), 391-393.<br />
|-<br />
|[KKW02] || '''S. Koshitani, N. Kunugi and K. Waki''', ''Broué's conjecture for non-principal 3-blocks of finite groups'', J. Pure and Applied Algebra '''173''' (2002), 177-211. <br />
|-<br />
|[KKW04] || '''S. Koshitani, N. Kunugi and K. Waki''', ''Broué's abelian defect group conjecture for Held group and the sporadic Suzuki group'', J. Algebra '''279''' (2004), 638-666. <br />
|-<br />
|[KoLa20] || '''S. Koshitani and C. Lassueur''', ''Splendid Morita equivalences for principal 2-blocks with dihedral defect groups'', Math. Z. '''294''' (2020), 639-666.<br />
|-<br />
|[KoLa20b] || '''S. Koshitani and C. Lassueur''', ''Splendid Morita equivalences for principal blocks with generalised quaternion defect groups'', J. Algebra '''558''' (2020), 523-533.<br />
|-<br />
|[KoLaSa22] || '''S. Koshitani, C. Lassueur and B. Sambale''', ''Splendid Morita equivalences for principal blocks with semidihedral defect groups'', Proceedings of the American Mathematical Society '''150''' (2022), 41-53.<br />
|-<br />
|[KoLaSa23] || '''S. Koshitani, C. Lassueur and B. Sambale''', [https://arxiv.org/abs/2310.13621 ''Principal <math>2</math>-blocks with wreathed defect groups up to splendid Morita equivalence''], [https://arxiv.org/abs/2310.13621 arxiv:2310.13621]<br />
|-<br />
|[Kü80] || '''B. Külshammer''', ''On 2-blocks with wreathed defect groups'', J. Algebra '''64''' (1980), 529–555.<br />
|-<br />
|[Kü81] || '''B. Külshammer''', ''On p-blocks of p-solvable groups'', Comm. Alg. '''9''' (1981), 1763-1785.<br />
|-<br />
|[Kü91] || '''B. Külshammer''', ''Group-theoretical descriptions of ring-theoretical invariants of group algebras'', in Representation Theory of Finite Groups and Finite-Dimensional Algebras (Bielefeld, 1991), Progr. Math. '''95''', pp. 425-442, Birkhauser (1991).<br />
|-<br />
|[Kü95] || '''B. Külshammer''', ''Donovan's conjecture, crossed products and algebraic group actions'', Israel J. Math. '''92''' (1995), 295-306.<br />
|-<br />
|[KS13] || '''B. Külshammer and B. Sambale''', ''The 2-blocks of defect 4'', Representation Theory '''17''' (2013), 226-236.<br />
|-<br />
|[Ku00] || '''N. Kunugi''', ''Morita equivalent 3-blocks of the 3-dimensional projective special linear groups'', Proc. LMS '''80''' (2000), 575-589.<br />
|-<br />
|[Kup69] || '''H. Kupisch''', ''Unzerlegbare Moduln endlicher Gruppen mit zyklischer p-Sylow Gruppe'', Math. Z. '''108''' (1969), 77-104.<br />
|- id="L"<br />
|[LM80]||'''P. Landrock and G. O. Michler''', ''Principal 2-blocks of the simple groups of Ree type'', Trans. AMS '''260''' (1980), 83-111.<br />
|-<br />
|[Li94] || '''M. Linckelmann''', ''The source algebras of blocks with a Klein four defect group'', J. Algebra '''167''' (1994), 821-854.<br />
|-<br />
|[Li94b] || '''M. Linckelmann''', ''A derived equivalence for blocks with dihedral defect groups'', J. Algebra '''164''' (1994), 244-255. <br />
|-<br />
|[Li96] || '''M. Linckelmann''', ''The isomorphism problem for cyclic blocks and their source algebras'', Invent. Math. '''125''' (1996), 265-283.<br />
|-<br />
|[Li18] || '''M. Linckelmann''', [https://arxiv.org/abs/1805.08884 ''The strong Frobenius numbers for cyclic defect blocks are equal to one''], [https://arxiv.org/abs/1805.08884 arXiv:1805.08884]<br />
|-<br />
|[Li18b] || '''M. Linckelmann''', ''Finite-dimensional algebras arising as blocks of finite group algebras'', Contemporary Mathematics '''705''' (2018), 155-188.<br />
|-<br />
|[Li18c] || '''M. Linckelmann''', ''The block theory of finite group algebras, Volume 1'', London Math. Soc. Student Texts '''92''', Cambridge University Press (2018).<br />
|-<br />
|[Li18d] || '''M. Linckelmann''', ''The block theory of finite group algebras, Volume 2'', London Math. Soc. Student Texts '''92''', Cambridge University Press (2018).<br />
|-<br />
|[LM20] || '''M. Linckelmann and W. Murphy''', [https://arxiv.org/abs/2005.02223 ''A 9-dimensional algebra which is not a block of a finite group''], Quarterly Journal of Mathematics 72 (2021), 1077–1088<br />
|-<br />
|[Liv19] || '''M. Livesey''', [https://arxiv.org/abs/1907.12167 ''On Picard groups of blocks with normal defect groups''], J. Algebra '''566''' (2021), 94-118.<br />
|-<br />
|[LiMa20] || '''M. Livesey and C. Marchi''', [https://arxiv.org/abs/2002.10571 ''On Picent for blocks with normal defect group''], [https://arxiv.org/abs/2002.10571 arXiv:2002.10571]<br />
|-<br />
|[LiMa20b] || '''M. Livesey and C. Marchi''', [https://arxiv.org/abs/2008.05857 ''Picard groups for blocks with normal defect groups and linear source bimodules''], [https://arxiv.org/abs/2008.05857 arXiv:2008.05857]<br />
|- id="M"<br />
|[Mac] || '''N. Macgregor''', ''Morita equivalence classes of tame blocks of finite groups'', J. Algebra '''608''' (2022), 719-754.<br />
|-<br />
|[Mar] || '''C. Marchi''', ''Picard groups for blocks'', PhD thesis, University of Manchester (2022)<br />
|-<br />
|[Ma86] || '''U. Martin''', ''Almost all <math>p</math>-groups have automorphism group a <math>p</math>-group'', Bull. AMS '''15''' (1986), 78-82.<br />
|-<br />
|[McK19] || '''E. McKernon''', [https://arxiv.org/abs/1912.03222 ''2-Blocks whose defect group is homocyclic and whose inertial quotient contains a Singer cycle''], J. Algebra '''563''' (2020), 30–48.<br />
|-<br />
|[MS08] || '''J. Müller and M. Schaps''', ''The Broué conjecture for the faithful 3-blocks of <math>4.M_{22}</math>'', J. Algebra '''319''' (2008), 3588-3602.<br />
|- id="N"<br />
|[NS18] || '''G. Navarro and B. Sambale''', ''On the blockwise modular isomorphism problem'', Manuscripta Math. '''157''' (2018), 263-278.<br />
|- <br />
|[Ne02] || '''G. Nebe''', [http://www.math.rwth-aachen.de/~Gabriele.Nebe/papers/survey.pdf ''Group rings of finite groups over p-adic integers, some examples''], Proceedings of the conference Around Group rings (Edmonton) Resenhas '''5''' (2002), 329-350.<br />
|- id="O"<br />
|[Ok97] || '''T. Okuyama''', ''Some examples of derived equivalent blocks of finite groups'', preprint (1997).<br />
|- id="P"<br />
|[Pu88]|| '''L. Puig''', ''Nilpotent blocks and their source algebras'', Invent. Math. '''93''' (1988), 77-116.<br />
|-<br />
|[Pu94] || '''L. Puig''', ''On Joanna Scopes’ criterion of equivalence for blocks of symmetric groups'', Algebra Colloq. '''1''' (1994), 25-55.<br />
|-<br />
|[Pu99]|| '''L. Puig''', ''On the local structure of Morita and Rickard equivalences between Brauer blocks'', Progress in Math. '''178''', Birkhauser Verlag (1999).<br />
|-<br />
|[Pu09] || '''L. Puig''', ''Block source algebras in p-solvable groups'', Michigan Math. J. '''58''' (2009), 323-338.<br />
|- id="R"<br />
|[Ri96] || '''J. Rickard''', ''Splendid equivalences: derived categories and permutation modules'', Proc. London Math. Soc. '''72''' (1996), 331-358.<br />
|-<br />
|[Ro95] || '''R. Rouquier''', ''From stable equivalences to Rickard equivalences for blocks with cyclic defect'', Proceedings of Groups 1993, Galway-St. Andrews Conference, Vol. 2, London Math. Soc. Lecture Note Ser. '''212''', Cambridge University Press (1995), 512-523.<br />
|-<br />
|[Ru11] || '''P. Ruengrot''', ''Perfect isometry groups for blocks of finite groups'', PhD Thesis, University of Manchester (2011).<br />
|- id="S"<br />
|[Sa11] || '''B. Sambale''', ''<math>2</math>-blocks with minimal nonabelian defect groups'', J. Algebra '''337''' (2011), 261–284.<br />
|-<br />
|[Sa12] || '''B. Sambale''', ''Blocks with defect group <math>D_{2^n} \times C_{2^m}</math>'', J. Pure Appl. Algebra '''216''' (2012), 119–125.<br />
|-<br />
|[Sa12b] || '''B. Sambale''', ''Fusion systems on metacyclic 2-groups'', Osaka J. Math. '''49''' (2012), 325–329.<br />
|-<br />
|[Sa13] || '''B. Sambale''', ''Blocks with defect group <math>Q_{2^n} \times C_{2^m}</math> and <math>SD_{2^n} \times C_{2^m}</math>'', Algebr. Represent. Theory '''16''' (2013), 1717–1732.<br />
|-<br />
|[Sa13b] || '''B. Sambale''', ''Blocks with central product defect group <math>D_{2^n} ∗ C_{2^m}</math>'', Proc. Amer. Math. Soc. '''141''' (2013), 4057–4069.<br />
|-<br />
|[Sa13c] || '''B. Sambale''', ''Further evidence for conjectures in block theory'', Algebra Number Theory '''7''' (2013), 2241–2273. <br />
|-<br />
|[Sa14] || '''B. Sambale''', ''Blocks of Finite Groups and Their Invariants'', Lecture Notes in Mathematics, Springer (2014).<br />
|-<br />
|[Sa16] || '''B. Sambale''', ''2-blocks with minimal nonabelian defect groups III'', Pacific J. Math. '''280''' (2016), 475–487.<br />
|-<br />
|[Sa20] || '''B. Sambale''', [https://arxiv.org/abs/2005.13172 ''Blocks with small-dimensional basic algebra''], Bul. Aust. Math. Soc. '''103''' (2021), 461-474.<br />
|-<br />
|[SSS98] || '''M. Schaps, D. Shapira and O. Shlomo''', ''Quivers of blocks with normal defect groups'', Proc. Symp. in Pure Mathematics '''63''', Amer. Math. Soc. (1998), 497-510.<br />
|-<br />
|[Sc91] || '''J. Scopes''', ''Cartan matrices and Morita equivalence for blocks of the symmetric groups'', J. Algebra '''142''' (1991), 441-455.<br />
|-<br />
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|-<br />
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|- id="T"<br />
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|- id="V"<br />
|[vdW91] || '''R. van der Waall''', ''On p-nilpotent forcing groups'', Indag. Mathem., N.S., '''2''' (1991), 367-384.<br />
|- id="W"<br />
|[Wa00]|| '''A. Watanabe''', ''A remark on a splitting theorem for blocks with abelian defect groups'', RIMS Kokyuroku '''1140''', Edited by H.Sasaki, Research Institute for Mathematical Sciences, Kyoto University (2000), 76-79.<br />
|-<br />
|[WZZ18] || '''Chao Wu, Kun Zhang and Yuanyang Zhou''', ''[https://arxiv.org/abs/1803.07262 Blocks with defect group <math>Z_{2^n} \times Z_{2^n} \times Z_{2^m}</math>]'', J. Algebra '''510''' (2018), 469-498.<br />
|}</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Guide_to_contributing:_Code&diff=1241
Guide to contributing: Code
2024-01-10T19:08:41Z
<p>Charles Eaton: Removed GAP code heading</p>
<hr />
<div>The code in this section computes the Cartan matrix, the decomposition matrix and the Loewy structure of the projective indecomposable modules. The output is formatted in LaTeX, so it can be copied and pasted easily on the relevant block page.<br />
<br />
== Magma code ==<br />
<br />
The following code, written by Claudio Marchi and [[User:CesareGArdito|Cesare G. Ardito]], can usually be run on the free Magma online calculator [http://magma.maths.usyd.edu.au/calc/], as long as the involved group is small.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="overflow:auto;"><br />
<div style="font-weight:bold;line-height:1.6;">Magma code</div><br />
<div class="mw-collapsible-content"><br />
<pre><br />
<br />
/* Input data: a group, a prime and an exponent such that GF(p^n) contains all the necessary roots of unity */<br />
G:=SmallGroup(2,1); <br />
p:=2;<br />
splittingexponent:=1;<br />
/* End of input data*/<br />
<br />
LoewyStructure:= function(W,p,k,S,irrs)<br />
Z:=W;<br />
t:=0;<br />
maxlength:=100;<br />
M:=ZeroMatrix(Integers(),maxlength,maxlength);<br />
repeat <br />
t:=t+1;<br />
J:=JacobsonRadical(W);<br />
V:=W/J;<br />
V:=ChangeRing(V,GF(p^k));<br />
if IsDecomposable(V) then<br />
indsum:=IndecomposableSummands(V);<br />
for i in [1..#indsum] do<br />
for j in S do<br />
V7:=indsum[i];<br />
U:=irrs[j];<br />
U:=ChangeRing(U,GF(p^k));<br />
V:=ChangeRing(V7,GF(p^k));<br />
if IsIsomorphic(V,U) then<br />
M[t][i]:=j; <br />
continue i;<br />
end if;<br />
end for;<br />
end for; <br />
else<br />
for j in S do<br />
U:=irrs[j];<br />
U:=ChangeRing(U,GF(p^k));<br />
V:=ChangeRing(V,GF(p^k));<br />
if IsIsomorphic(V,U) then<br />
M[t][1]:=j;<br />
break j;<br />
end if;<br />
end for;<br />
end if; <br />
W:=J;<br />
if IsIrreducible(W) then<br />
t:=t+1;<br />
for j in S do<br />
U:=irrs[j];<br />
U:=ChangeRing(U,GF(p^k));<br />
V:=ChangeRing(W,GF(p^k));<br />
if IsIsomorphic(V,U) then<br />
M[t][1]:=j;<br />
break j;<br />
end if;<br />
end for; <br />
end if;<br />
until IsIrreducible(W);<br />
printf("\n");<br />
for i in [1..maxlength] do<br />
flag:=0;<br />
for j in [1..maxlength] do<br />
if(not (M[i][j] eq 0)) then <br />
flag:=1;<br />
printf "S_\{%o\} ", M[i][j]; <br />
end if;<br />
end for;<br />
if(flag eq 1) then<br />
printf("\\\\ \n");<br />
end if;<br />
end for;<br />
print "The structure above is the Loewy structure of";<br />
return Z;<br />
end function;<br />
<br />
PrintLatex:= function(M)<br />
for i in [1..NumberOfRows(M)-1] do<br />
for j in [1..NumberOfColumns(M)-1] do<br />
printf "%o & ", M[i,j];<br />
end for;<br />
printf "%o \\\\ \n", M[i,NumberOfColumns(M)];<br />
end for;<br />
for j in [1..NumberOfColumns(M)-1] do<br />
printf "%o & ", M[NumberOfRows(M),j];<br />
end for;<br />
printf "%o \n", M[NumberOfRows(M),NumberOfColumns(M)];<br />
return " ";<br />
end function;<br />
<br />
PrintLatex(AbsoluteCartanMatrix(G,GF(p^splittingexponent)));<br />
printf("\n");<br />
PrintLatex(DecompositionMatrix(G,GF(p^splittingexponent)));<br />
irrs:=AbsolutelyIrreducibleModules(G, GF(p^splittingexponent));<br />
P:=ProjectiveIndecomposableModules(G, GF(p^splittingexponent));<br />
for i in [1..#P] do<br />
LoewyStructure(P[i],p,splittingexponent,[1..#irrs],irrs);<br />
end for;<br />
</pre><br />
</div></div><br />
Note that, if <math>G</math> has more than one block, some manual work on the output is needed to isolate the relevant submatrices and indecomposable projective modules that belong to the block that is being investigated.</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Classification_by_p-group&diff=1240
Classification by p-group
2024-01-10T18:59:52Z
<p>Charles Eaton: /* Blocks for p=2 */</p>
<hr />
<div>'''Classification of Morita equivalences for blocks with a given defect group'''<br />
<br />
On this page we list classifications of Morita equivalence classes for each isomorphism class of p-groups in turn. [[Generic classifications by p-group class|Generic classifications for classes of p-groups can be found here]].<br />
<br />
See [[Labelling for Morita equivalence classes|this page]] for a description of the labelling conventions.<br />
<br />
== Blocks for <math> p=2 </math> ==<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>1 \leq |D| \leq 8</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|- <br />
| 2 || [[C2|1]] || [[C2|<math>C_2</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
| 4 || [[C4|1]] || [[C4|<math>C_4</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
| 4 || [[C2xC2|2]] || [[C2xC2|<math>C_2 \times C_2</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Er82], [Li94] ]] ||<br />
|- <br />
|8 || [[C8|1]] || [[C8|<math>C_8</math>]] ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|8 || [[C4xC2|2]] || [[C4xC2|<math>C_4 \times C_2</math>]] ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|8 || [[D8|3]] || [[D8|<math>D_8</math>]] ||6(?) || <math>k</math> || <math>k</math> || [[References|[Er87] ]] || Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] <br />
|-<br />
|8 || [[Q8|4]] || [[Q8|<math>Q_8</math>]] ||3(3) || <math>\mathcal{O}</math> || <math>k</math> || [[References|[Er88a], [Er88b], [HKL07], [Ei16]]] || <br />
|-<br />
|8 || [[C2xC2xC2|5]] || [[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]] || 8(8) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References| [Ea16]]] || Uses CFSG<br />
|}<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>|D|=16</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Donovan (w.r.t)?<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|16 || [[C16|1]] || [[C16|<math>C_{16}</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|16 || [[C4xC4|2]] || [[C4xC4|<math>C_4 \times C_4</math>]] || <math>\mathcal{O}</math> || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[EKKS14] ]] ||<br />
|-<br />
|16 || [[MNA(2,1)|3]] || [[MNA(2,1)]] || No || 3(?) || No || || [[References|[Sa11] ]] || Block invariants known<br />
|-<br />
|16 || [[C4:C4|4]] || [[C4:C4|<math>C_4:C_4</math>]] || <math>\mathcal{O}</math>|| 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[CG12], [Sa12b]]] ||<br />
|-<br />
|16 || [[C8xC2|5]] || [[C8xC2|<math>C_8 \times C_2</math>]]|| <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|16 || [[M16|6]] || [[M16|<math>M_{16}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[CG12], [Sa12b] ]] || <br />
|-<br />
|16 || [[D16|7]] || [[D16|<math>D_{16}</math>]] || <math>k</math>|| 5(?) || <math>k</math> || <math>k</math> || [[References|[Er87]]] || Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] <br />
|-<br />
|16 || [[SD16|8]] || [[SD16|<math>SD_{16}</math>]] || <math>k</math> || 7(?) || || || [[References|[Er88c], [Er90b]]] || Two other possible classes<br />
|-<br />
|16 || [[Q16|9]] || [[Q16|<math>Q_{16}</math>]] || No || 6(?) || || <math>k</math> || [[References|[Er88a], [Er88b], [Ho97]]] || Two possibly infinite families when <math>l(B)=2</math>. Classified over <math>\mathcal{O}</math> when <math>l(B)=3</math> in [[References#E|[Ei16]]]. Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20b]]]<br />
|-<br />
|16 || [[C4xC2xC2|10]] || [[C4xC2xC2|<math>C_4 \times C_2 \times C_2</math>]]|| <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[EL18a]]] ||<br />
|-<br />
|16 || [[D8xC2|11]] || [[D8xC2|<math>D_8 \times C_2</math>]] || No || || || || [[References|[Sa12] ]] || Block invariants known<br />
|-<br />
|16 || [[Q8xC2|12]] || [[Q8xC2|<math>Q_8 \times C_2</math>]] || <math>\mathcal{O}</math> || 3(3) || No || || [[References#E|[EL20]]] || Block invariants known by [[References#S|[Sa13]]]<br />
|-<br />
|16 || [[D8*C4|13]] || [[D8*C4|<math>D_8*C_4</math>]] || No || 3(?) || No || || [[References|[Sa13b] ]] || Block invariants known<br />
|-<br />
|16 || [[(C2)^4|14]] || [[(C2)^4|<math>(C_2)^4</math>]] || <math>\mathcal{O}</math> || 16(16) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Ea18] ]] ||<br />
|}<br />
<br />
The table for defect groups of order 32 takes as its starting point Table 13.1 of Sambale's book [[References|[Sa14]]].<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>|D|=32</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Donovan (w.r.t)?<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|32 || [[C32|1]] || [[C32|<math>C_{32}</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|32 || [[MNA(2,2)|2]] || [[MNA(2,2)|<math>MNA(2,2)</math>]] || <math>\mathcal{O}</math> || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EKS12]]] ||<br />
|-<br />
|32 || [[C8xC4|3]] || [[C8xC4|<math>C_8 \times C_4</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|-<br />
|32 || [[C8:C4|4]] || [[C8:C4|<math>C_8:C_4</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[MNA(3,1)|5]] || [[MNA(3,1)|<math>MNA(3,1)</math>]] || No || || || || [[References#S|[Sa11] ]] || Invariants known<br />
|-<br />
|32 || [[MNA(2,1):C2|6]] || [[MNA(3,1):C2|<math>MNA(2,1):C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,7)|7]] || [[SmallGroup(32,7)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] || <math>M_{16}:C_2</math><br />
|-<br />
|32 || [[2.MNA(2,1)|8]] || [[2.MNA(2,1)|<math>2.MNA(2,1)</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[D8:C4|9]] || [[D8:C4|<math>D_8:C_4</math>]] || No || || || || [[References#S|[Sa14,10.23]]] || Invariants known<br />
|-<br />
|32 || [[Q8:C4|10]] || [[Q8:C4|<math>Q_8:C_4</math>]] || No || || || || [[References#S|[Sa14,10.25]]] || Invariants known<br />
|-<br />
|32 || [[C4wrC2|11]] || [[C4wrC2|<math>C_4 \wr C_2</math>]] || No || 6(6) || No || || [[References#K|[Ku80]]], [[References#K|[KoLaSa23]]] || Invariants known. Principal blocks classified up to source algebra equivalence in [[References#K|[KoLaSa23]]]<br />
|-<br />
|32 || [[C4:C8|12]] || [[C4:C8|<math>C_4:C_8</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[C8:C4a|13]] || [[C8:C4a|<math>C_8:C_4=\langle a,b|a^8=b^4=1, ba=a^3b \rangle</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[C8:C4b|14]] || [[C8:C4b|<math>C_8:C_4=\langle a,b|a^8=b^4=1, ba=a^7b \rangle</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,15)|15]] || [[SmallGroup(32,15)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[C16xC2|16]] || [[C16xC2|<math>C_{16} \times C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|32 || [[M32|17]] || [[M32|<math>M_{32}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[D32|18]] || [[D32|<math>D_{32}</math>]] || <math>k</math> || 5(?) || <math>k</math> || <math>k</math> || [[References#E|[Er87]]] || Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] <br />
|-<br />
|32 || [[SD32|19]] || [[SD32|<math>SD_{32}</math>]] || <math>k</math> || || || || ||<br />
|-<br />
|32 || [[Q32|20]] || [[Q32|<math>Q_{32}</math>]] || No || || || || [[References#E|[Er88a], [Er88b], [Ho97]]] || Two possibly infinite families when <math>l(B)=2</math>. Classified over <math>\mathcal{O}</math> when <math>l(B)=3</math> in [[References#E|[Ei16]]]. Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20b]]]<br />
|-<br />
|32 || [[C4xC4xC2|21]] || [[C4xC4xC2|<math>C_4 \times C_4 \times C_2</math>]] || <math>\mathcal{O}</math> || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EKKS14]]] <br />
|-<br />
|32 || [[MNA(2,1)xC2|22]] || [[MNA(2,1)xC2|<math>MNA(2,1) \times C_2</math>]] || No || || || || [[References#S|[Sa14,10.25]]] || Invariants known<br />
|-<br />
|32 || [[(C4:C4)xC2|23]] || [[(C4:C4)xC2|<math>(C_4:C_4) \times C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,24)|24]] || [[SmallGroup(32,24)]]<!--<math>(C_4 \times C_4):C_2=\langle a,b,c \mid a^4 = b^4 = c^2 = e, ab = ba, ac = ca, cb = a^2bc \rangle</math>]]--> || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[D8xC4|25]] || [[D8xC4|<math>D_8 \times C_4</math>]] || No || || || || [[References#S|[Sa14,9.7]]] ||<br />
Invariants known<br />
|-<br />
|32 || [[Q8xC4|26]] || [[Q8xC4|<math>Q_8 \times C_4</math>]] || <math>\mathcal{O}</math> || 3(3) || No || || [[References#E|[EL20]]] || Invariants known by [[References#S|[Sa14,9.28]]]<br />
|-<br />
|32 || [[SmallGroup(32,27)|27]] || [[SmallGroup(32,27)]]<!--|<math>(C_4 \times C_4):C_2=\langle x,y,z,a,b \mid x^2 = y^2 = z^2 = a^2 = b^2 = e, xy = yx, xz, = zx, yz = zy, aza^{-1} = xz, bzb^{-1} = yz, ax = xa, ay = ya, bx = xb, by = yb \rangle</math>]]--> || No || || || || ||<br />
|-<br />
|32 || [[SmallGroup(32,28)|28]] || [[SmallGroup(32,28)]] || No || || || || [[References#S|[Sa14,13.11]]] || Invariants known<br />
|-<br />
|32 || [[SmallGroup(32,29)|29]] || [[SmallGroup(32,29)]] || No || || || || [[References#S|[Sa14,13.11]]] || Invariants known<br />
|-<br />
|32 || [[SmallGroup(32,30)|30]] || [[SmallGroup(32,30)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,31)|31]] || [[SmallGroup(32,31)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,32)|32]] || [[SmallGroup(32,32)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,33)|33]] || [[SmallGroup(32,33)]] || No || || || || [[References|[Sa14,13.12]]] || Invariants partly known<br />
|-<br />
|32 || [[SmallGroup(32,34)|34]] || [[SmallGroup(32,34)]] || No || || || || [[References|[Sa14,13.12]]] || Invariants partly known<br />
|-<br />
|32 || [[C4:Q8|35]] || [[C4:Q8|<math>C_4:Q_8</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[C8xC2xC2|36]] || [[C8xC2xC2|<math>C_8 \times C_2 \times C_2</math>]] || <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EL18a]]] ||<br />
|-<br />
|32 || [[M16xC2|37]] || [[M16xC2|<math>M_{16} \times C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[D8*C8|38]] || [[D8*C8|<math>D_8*C_8</math>]] || No || || || || [[References#S|[Sa14,9.18]]] || Invariants known<br />
|-<br />
|32 || [[D16xC2|39]] || [[D16xC2|<math>D_{16} \times C_2</math>]] || No || || || || [[References#S|[Sa14,9.7]]] || Invariants known<br />
|-<br />
|32 || [[SD16xC2|40]] || [[SD16xC2|<math>SD_{16} \times C_2</math>]] || No || || || || [[References#S|[Sa14,9.37]]] || Invariants known<br />
|-<br />
|32 || [[Q16xC2|41]] || [[Q16xC2|<math>Q_{16} \times C_2</math>]] || No || || || || [[References#S|[Sa14,9.28]]] || Invariants known<br />
|-<br />
|32 || [[D16*C4|42]] || [[D16*C4|<math>D_{16}*C_4</math>]] || No || || || || [[References|[Sa14,9.18]]] || Invariants known<br />
|-<br />
|32 || [[SmallGroup(32,43)|43]] || [[SmallGroup(32,43)]] || No || || || || ||<br />
|-<br />
|32 || [[SmallGroup(32,44)|44]] || [[SmallGroup(32,44)]] || No || || || || ||<br />
|-<br />
|32 || [[C4xC2xC2xC2|45]] || [[C4xC2xC2xC2|<math>C_4 \times C_2 \times C_2 \times C_2</math>]] || <math>\mathcal{O}</math> || 8(8) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EL23]]] ||<br />
|-<br />
|32 || [[D8xC2xC2|46]] || [[D8xC2xC2|<math>D_8 \times C_2 \times C_2</math>]] || No || || || || ||<br />
|-<br />
|32 || [[Q8xC2xC2|47]] || [[Q8xC2xC2|<math>Q_8 \times C_2 \times C_2</math>]] || No || || || || ||<br />
|-<br />
|32 || [[D8*C4xC2|48]] || [[D8*C4xC2|<math>(D_8*C_4) \times C_2</math>]] || No || || || || ||<br />
|-<br />
|32 || [[D8*D8|49]] || [[D8*D8|<math>D_8*D_8</math>]] || No || || || || [[References#S|[Sa13c]]] || Invariants partly known<br />
|-<br />
|32 || [[D8*Q8|50]] || [[D8*Q8|<math>D_8*Q_8</math>]] || No || || || || [[References#S|[Sa13c]]] || Invariants partly known<br />
|-<br />
|32 || [[(C2)^5|51]] || [[(C2)^5|<math>(C_2)^5</math>]] || <math>\mathcal{O}</math> || 34 (34) || <math>\mathcal{O}</math> || || [[References#A|[Ar19]]] || Derived eq. classes determined for 30 of the 34 Morita eq. classes. <br />
|}<br />
<br />
<!--<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>|D|=64</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Donovan (w.r.t)?<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|64 || [[C64|1]] || [[C64|<math>C_{64}</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
|64 || [[C8xC8|2]] || [[C8xC8|<math>C_8 \times C_8</math>]] || <math>\mathcal{O}</math> ||2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EKKS14]]]||<br />
|-<br />
|64 || [[SmallGroup(64,3)|3]] || [[SmallGroup(64,3)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_8:C_8</math><br />
|-<br />
|64 || [[(C2xC2xC2):C8|4]] || [[(C2xC2xC2):C8|<math>(C_2)^3:C_8</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,5)|5]] || [[SmallGroup(64,5)]] || No || || || || ||<br />
|-<br />
|64 || [[(D8:C8|6]] || [[D8:C8|<math>D_8:C_8</math>]] || No || || || || ||<br />
|-<br />
|64 || [[(Q8:C8|7]] || [[Q8:C8|<math>Q_8:C_8</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,8)|8]] || [[SmallGroup(64,8)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,9)|9]] || [[SmallGroup(64,9)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,10)|10]] || [[SmallGroup(64,10)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,11)|11]] || [[SmallGroup(64,11)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,12)|12]] || [[SmallGroup(64,12)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,13)|13]] || [[SmallGroup(64,13)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,14)|14]] || [[SmallGroup(64,14)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,15)|15]] || [[SmallGroup(64,15)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_8:C_8</math><br />
|-<br />
|64 || [[SmallGroup(64,16)|16]] || [[SmallGroup(64,16)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_8:C_8</math><br />
|-<br />
|64 || [[SmallGroup(64,17)|17]] || [[SmallGroup(64,17)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,18)|18]] || [[SmallGroup(64,18)]] || No || || || || || <math>(C_4 \times C_4):C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,19)|19]] || [[SmallGroup(64,19)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,20)|20]] || [[SmallGroup(64,20)]] || No || || || || || <math>(C_4 \times C_4):C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,21)|21]] || [[SmallGroup(64,21)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,22)|22]] || [[SmallGroup(64,22)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,23)|23]] || [[SmallGroup(64,23)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,24)|24]] || [[SmallGroup(64,24)]] || No || || || || || <math>M_{16}:C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,25)|25]] || [[SmallGroup(64,25)]] || No || || || || || <math>M_{16}:C_4</math><br />
|-<br />
|64 || [[C16xC4|26]] || [[C16xC4|<math>C_{16} \times C_4</math>]]|| <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|64 || [[SmallGroup(64,27)|27]] || [[SmallGroup(64,27)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_{16}:C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,28)|28]] || [[SmallGroup(64,28)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_{16}:C_4</math><br />
|-<br />
|64 || [[(C2xC2):C16|29]] || [[(C2xC2):C16|<math>(C_2)^2:C_{16}</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,30)|30]] || [[SmallGroup(64,30)]] || No || || || || || <math>M_{32}:C_2</math><br />
|-<br />
|64 || [[SmallGroup(64,31)|31]] || [[SmallGroup(64,31)]] || No || || || || || <math>M_{32}:C_2</math><br />
|-<br />
|64 || [[C2wrC4|32]] || [[(C2wrC4|<math>C_2 \wr C_4</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,33)|33]] || [[SmallGroup(64,33)]] || No || || || || || <br />
|-<br />
|64 || [[SmallGroup(64,34)|34]] || [[SmallGroup(64,31)]] || No || || || || || <math>(C_4 \times C_4):C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,35)|35]] || [[SmallGroup(64,35)]] || No || || || || || <math>(C_4 \times C_4):C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,36)|36]] || [[SmallGroup(64,36)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,37)|37]] || [[SmallGroup(64,37)]] || No || || || || || <math>C_4:Q_8</math>, fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,38)|38]] || [[SmallGroup(64,38)]] || No || || || || || <math>D_{16}:C_4</math>, fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,39)|39]] || [[SmallGroup(64,39)]] || No || || || || || <math>Q_{16}:C_2</math><br />
|-<br />
|64 || [[SmallGroup(64,40)|40]] || [[SmallGroup(64,40)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,41)|41]] || [[SmallGroup(64,41)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,42)|42]] || [[SmallGroup(64,42)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,43)|43]] || [[SmallGroup(64,43)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[C4:C16|44]] || [[C4:C16|<math>C_4:C_{16}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_4:C_{16}</math><br />
|-<br />
|64 || [[SmallGroup(64,45)|45]] || [[SmallGroup(64,45)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,46)|46]] || [[SmallGroup(64,46)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,47)|47]] || [[SmallGroup(64,47)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_{16}:C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,48)|48]] || [[SmallGroup(64,48)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_{16}:C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,48)|49]] || [[SmallGroup(64,49)]] || No || || || || ||<br />
|- <br />
|64 || [[C32xC2|50]] || [[C32xC2|<math>C_{32} \times C_2</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|64 || [[M6(2)|51]] || [[M6(2)|<math>M_6(2)</math>]] || No || || || || ||<br />
|-<br />
|64 || [[D64|52]] || [[D64|<math>D_{64}</math>]] || No || <math>k</math> || || || ||<br />
|-<br />
|64 || [[SD64|53]] || [[SD64|<math>SD_{64}</math>]] || No || <math>k</math> || || || ||<br />
|-<br />
|64 || [[Q64|54]] || [[Q64|<math>Q_{64}</math>]] || No || || || || ||<br />
|- <br />
|64 || [[C4xC4xC4|55]] || [[C4xC4xC4|<math>C_4 \times C_4 \times C_4</math>]] || <math>\mathcal{O}</math> ||4(4) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EL18a]]]||<br />
|-<br />
|64 || [[SmallGroup(64,56)|56]] || [[SmallGroup(64,56)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,57)|57]] || [[SmallGroup(64,57)]] || No || || || || ||<br />
|- <br />
|64 || [[C4x(C2xC2):C4|58]] || [[C4x(C2xC2):C4|<math>C_{4} \times (C_2 \times C_2):C_4</math>]] || || || || || || Fusion trivial?<br />
|- <br />
|64 || [[C4x(C4:C4)|59]] || [[C4x(C4:C4)|<math>C_{4} \times (C_4:C_4)</math>]] || || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,60)|60]] || [[SmallGroup(64,60)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,61)|61]] || [[SmallGroup(64,61)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,62)|62]] || [[SmallGroup(64,62)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,63)|63]] || [[SmallGroup(64,63)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|-<br />
|64 || [[SmallGroup(64,64)|64]] || [[SmallGroup(64,64)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,65)|65]] || [[SmallGroup(64,65)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,66)|66]] || [[SmallGroup(64,66)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,67)|67]] || [[SmallGroup(64,67)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,68)|68]] || [[SmallGroup(64,68)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|-<br />
|64 || [[SmallGroup(64,69)|69]] || [[SmallGroup(64,69)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,70)|70]] || [[SmallGroup(64,70)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|-<br />
|64 || [[SmallGroup(64,71)|71]] || [[SmallGroup(64,71)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,72)|72]] || [[SmallGroup(64,72)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|-<br />
|64 || [[SmallGroup(64,73)|73]] || [[SmallGroup(64,73)]] || No || || || || ||<br />
|-<br />
|64 || [[(C2)^3:Q8|74]] || [[(C2)^3:Q8|<math>(C_2)^3:Q_8</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,75)|75]] || [[SmallGroup(64,75)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,76)|76]] || [[SmallGroup(64,76)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|-<br />
|64 || [[SmallGroup(64,77)|77]] || [[SmallGroup(64,77)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,78)|78]] || [[SmallGroup(64,78)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,79)|79]] || [[SmallGroup(64,79)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|-<br />
|64 || [[SmallGroup(64,80)|80]] || [[SmallGroup(64,80)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,81)|81]] || [[SmallGroup(64,81)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|-<br />
|64 || [[SmallGroup(64,82)|82]] || [[SmallGroup(64,82)]] || <math>\mathcal{O}</math> || 6(6) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[Ea24]]] || Sylow 2-subgroup of <math>Sz(8)</math><br />
|-<br />
|64 || [[C8xC4xC2|83]] || [[C8xC4xC2|<math>C_{8} \times C_4 \times C_2</math>]]|| <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|64 || [[C2x(C8:C4)|84]] || [[C2x(C8:C4)|<math>C_{2} \times (C_8:C_4)</math>]]|| No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[M4(2)xC4|85]] || [[M4(2)xC4|<math>M_4(2) \times C_4</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,86)|86]] || [[SmallGroup(64,86)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[C2x(C2xC2):C8|87]] || [[C2x(C2xC2):C8|<math>C_{2} \times (C_2 \times C_2):C_8</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,88)|88]] || [[SmallGroup(64,88)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[(C8xC2xC2):C2|89]] || [[(C8xC2xC2):C2|<math>(C_8 \times C_2 \times C_2):C_2</math>]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[C2x(C2xC2xC2):C4|90]] || [[C2x(C2xC2xC2):C4|<math>C_2 \times (C_2 \times C_2 \times C_2):C_4</math>]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,91)|91]] || [[SmallGroup(64,91)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,92)|92]] || [[SmallGroup(64,92)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,93)|93]] || [[SmallGroup(64,93)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,94)|94]] || [[SmallGroup(64,94)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[C2x(D_8:C4)|95]] || [[C2x(D_8:C4)|<math>C_{2} \times (D_8:C_4)</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[C2x(Q_8:C4)|96]] || [[C2x(Q_8:C4)|<math>C_{2} \times (Q_8:C_4)</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,97)|97]] || [[SmallGroup(64,97)]] || No || || || || || <br />
|-<br />
|64 || [[SmallGroup(64,98)|98]] || [[SmallGroup(64,98)]] || No || || || || || <br />
|-<br />
|64 || [[SmallGroup(64,99)|99]] || [[SmallGroup(64,99)]] || No || || || || || <br />
|-<br />
|64 || [[SmallGroup(64,100)|100]] || [[SmallGroup(64,100)]] || No || || || || || <br />
|-<br />
|64 || [[C2x(C4wrC2)|101]] || [[C2x(C4wrC2)|<math>C_{2} \times (C_4 \wr C_2)</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[(C4xC4)(C2:C2)|102]] || [[(C4xC4)(C2:C2)|<math>(C_4 \times C_4):(C_2 \times C_2)</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[C2x(C4:C8)|103]] || [[C2x(C4:C8)|<math>C_{2} \times (C_4:C_8)</math>]]|| No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[C4:M4(2)|104]] || [[C4:M4(2)|<math>C_{4}:M_4(2)</math>]]|| No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,105)|105]] || [[SmallGroup(64,105)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,106)|106]] || [[SmallGroup(64,106)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,107)|107]] || [[SmallGroup(64,107)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,108)|108]] || [[SmallGroup(64,108)]] || No || || || || ||<br />
|-<br />
|64 || [[M4(2):C4|109]] || [[M4(2):C4|<math>M_4(2):C_4</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,110)|110]] || [[SmallGroup(64,110)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,111)|111]] || [[SmallGroup(64,111)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,112)|112]] || [[SmallGroup(64,112)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|-<br />
|64 || [[SmallGroup(64,113)|113]] || [[SmallGroup(64,113)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,114)|114]] || [[SmallGroup(64,114)]] || No || || || || ||<br />
|-<br />
|64 || [[D8xC8|115]] || [[D8xC8|<math>D_8 \times C_8</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,116)|116]] || [[SmallGroup(64,116)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,117)|117]] || [[SmallGroup(64,117)]] || No || || || || ||<br />
|-<br />
|64 || [[D16xC4|118]] || [[D16xC4|<math>D_{16} \times C_4</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[SD16xC4|119]] || [[SD16xC4|<math>SD_{16} \times C_4</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[Q16xC4|120]] || [[Q16xC4|<math>Q_{16} \times C_4</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[SD16:C4|121]] || [[SD16:C4|<math>SD_{16}:C_4</math>]]|| No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[Q16:C4|122]] || [[Q16:C4|<math>Q_{16}:C_4</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[D16:C4|123]] || [[D16:C4|<math>D_{16}:C_4</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,124)|124]] || [[SmallGroup(64,124)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,125)|125]] || [[SmallGroup(64,125)]] || No || || || || ||<br />
|-<br />
|64 || [[Q8xC8|126]] || [[Q8xC8|<math>Q_{8} \times C_8</math>]]|| <math>\mathcal{O}</math> || 3(3) || || || [[References#E|[EL20]]] || Invariants known by [[References#S|[Sa14,9.28]]]<br />
|-<br />
|64 || [[SmallGroup(64,127)|127]] || [[SmallGroup(64,127)]] || No || || || || ||<br />
|-<br />
|64 || [[(C2xC2):D16|128]] || [[(C2xC2)D16|<math>(C_2 \times C_2):D_{16}</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[Q8:D8|129]] || [[Q8:D8|<math>Q_8:D_{8}</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[D8:D8|130]] || [[D8:D8|<math>D_8:D_{8}</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[Q8xQ8|239]] || [[Q8xQ8|<math>Q_{8} \times Q_8</math>]]|| <math>\mathcal{O}</math> || || || || [[References#E|[EL20]]] || <br />
|-<br />
|64 || [[SmallGroup(64,245)|245]] || [[SmallGroup(64,245)]] || <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[Ea24]]] || Sylow 2-subgroup of <math>PSU_3(4)</math><br />
<br />
|}--><br />
<br />
==Blocks for <math>p=3</math>==<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>1 \leq |D| \leq 27</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|- <br />
| 3 || [[C3|1]] || [[C3|<math>C_3</math>]] || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
|9 || [[C9|1]] ||[[C9|<math>C_9</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|-<br />
|9 || [[C3xC3|2]] || [[C3xC3|<math>C_3 \times C_3</math>]] || || || || ||<br />
|-<br />
|27 || [[C27|1]] || [[C27|<math>C_{27}</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|27 || [[C9xC3|2]] || [[C9xC3|<math>C_9 \times C_3</math>]] || || || || ||<br />
|-<br />
|27 || [[3_+^3|3]] || [[3_+^3|<math>3_+^{1+2}</math>]] || || || || ||<br />
|-<br />
|27 || [[3_-^3|4]] || [[3_-^3|<math>3_-^{1+2}</math>]] || || || || ||<br />
|-<br />
|27 || [[C3xC3xC3|5]] || [[C3xC3xC3|<math>C_3 \times C_3 \times C_3</math>]] || || || || ||<br />
|}<br />
<br />
==Blocks for <math>p=5</math>==<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>5 \leq |D| \leq 125</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|- <br />
|5 || [[C5|1]] || [[C5|<math>C_5</math>]] ||6(6) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
|25 || [[C25|1]] ||[[C25|<math>C_{25}</math>]] || 6(6) || No || <math>\mathcal{O}</math> || || Max 12 classes <br />
|-<br />
|25 || [[C5xC5|2]] || [[C5xC5|<math>C_5 \times C_5</math>]] || || || || ||<br />
|- <br />
|125 || [[C125|1]] ||[[C125|<math>C_{125}</math>]] || || || || || <br />
|-<br />
|125 || [[C25xC5|2]] || [[C25xC5|<math>C_{25} \times C_5</math>]] || || || || || <br />
|-<br />
|125 || [[5_+^3|3]] || [[5_+^3|<math>5_+^{1+2}</math>]] || 62(62) || <math>\mathcal{O}</math> || || [[References#A|[AE23]]] || Inertial quotients are consistent within classes<br />
|-<br />
|125 || [[5_-^3|4]] || [[5_-^3|<math>5_-^{1+2}</math>]] || || || || ||<br />
|-<br />
|125 || [[C5xC5xC5|5]] || [[C5xC5xC5|<math>C_5 \times C_5 \times C_5</math>]] || || || || ||<br />
|}<br />
<br />
==Blocks for <math>p\geq 7</math>==<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>|D|</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|- <br />
|7 || [[C7|1]] || [[C7|<math>C_7</math>]] ||14(14) ||No || <math>\mathcal{O}</math> || ||Max 21 classes <br />
|- <br />
|11|| [[C11|1]] || [[C11|<math>C_{11}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|- <br />
|13 || [[C13|1]] || [[C13|<math>C_{13}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|-<br />
|17|| [[C17|1]] || [[C17|<math>C_{17}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|- <br />
|19 || [[C19|1]] || [[C19|<math>C_{19}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|-<br />
|23 || [[C23|1]] || [[C23|<math>C_{23}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|}</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Classification_by_p-group&diff=1239
Classification by p-group
2024-01-10T14:31:50Z
<p>Charles Eaton: /* Blocks for p=2 */</p>
<hr />
<div>'''Classification of Morita equivalences for blocks with a given defect group'''<br />
<br />
On this page we list classifications of Morita equivalence classes for each isomorphism class of p-groups in turn. [[Generic classifications by p-group class|Generic classifications for classes of p-groups can be found here]].<br />
<br />
See [[Labelling for Morita equivalence classes|this page]] for a description of the labelling conventions.<br />
<br />
== Blocks for <math> p=2 </math> ==<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>1 \leq |D| \leq 8</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|- <br />
| 2 || [[C2|1]] || [[C2|<math>C_2</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
| 4 || [[C4|1]] || [[C4|<math>C_4</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
| 4 || [[C2xC2|2]] || [[C2xC2|<math>C_2 \times C_2</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Er82], [Li94] ]] ||<br />
|- <br />
|8 || [[C8|1]] || [[C8|<math>C_8</math>]] ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|8 || [[C4xC2|2]] || [[C4xC2|<math>C_4 \times C_2</math>]] ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|8 || [[D8|3]] || [[D8|<math>D_8</math>]] ||6(?) || <math>k</math> || <math>k</math> || [[References|[Er87] ]] || Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] <br />
|-<br />
|8 || [[Q8|4]] || [[Q8|<math>Q_8</math>]] ||3(3) || <math>\mathcal{O}</math> || <math>k</math> || [[References|[Er88a], [Er88b], [HKL07], [Ei16]]] || <br />
|-<br />
|8 || [[C2xC2xC2|5]] || [[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]] || 8(8) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References| [Ea16]]] || Uses CFSG<br />
|}<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>|D|=16</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Donovan (w.r.t)?<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|16 || [[C16|1]] || [[C16|<math>C_{16}</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|16 || [[C4xC4|2]] || [[C4xC4|<math>C_4 \times C_4</math>]] || <math>\mathcal{O}</math> || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[EKKS14] ]] ||<br />
|-<br />
|16 || [[MNA(2,1)|3]] || [[MNA(2,1)]] || No || 3(?) || No || || [[References|[Sa11] ]] || Block invariants known<br />
|-<br />
|16 || [[C4:C4|4]] || [[C4:C4|<math>C_4:C_4</math>]] || <math>\mathcal{O}</math>|| 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[CG12], [Sa12b]]] ||<br />
|-<br />
|16 || [[C8xC2|5]] || [[C8xC2|<math>C_8 \times C_2</math>]]|| <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|16 || [[M16|6]] || [[M16|<math>M_{16}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[CG12], [Sa12b] ]] || <br />
|-<br />
|16 || [[D16|7]] || [[D16|<math>D_{16}</math>]] || <math>k</math>|| 5(?) || <math>k</math> || <math>k</math> || [[References|[Er87]]] || Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] <br />
|-<br />
|16 || [[SD16|8]] || [[SD16|<math>SD_{16}</math>]] || <math>k</math> || 7(?) || || || [[References|[Er88c], [Er90b]]] || Two other possible classes<br />
|-<br />
|16 || [[Q16|9]] || [[Q16|<math>Q_{16}</math>]] || No || 6(?) || || <math>k</math> || [[References|[Er88a], [Er88b], [Ho97]]] || Two possibly infinite families when <math>l(B)=2</math>. Classified over <math>\mathcal{O}</math> when <math>l(B)=3</math> in [[References#E|[Ei16]]]. Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20b]]]<br />
|-<br />
|16 || [[C4xC2xC2|10]] || [[C4xC2xC2|<math>C_4 \times C_2 \times C_2</math>]]|| <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[EL18a]]] ||<br />
|-<br />
|16 || [[D8xC2|11]] || [[D8xC2|<math>D_8 \times C_2</math>]] || No || || || || [[References|[Sa12] ]] || Block invariants known<br />
|-<br />
|16 || [[Q8xC2|12]] || [[Q8xC2|<math>Q_8 \times C_2</math>]] || <math>\mathcal{O}</math> || 3(3) || No || || [[References#E|[EL20]]] || Block invariants known by [[References#S|[Sa13]]]<br />
|-<br />
|16 || [[D8*C4|13]] || [[D8*C4|<math>D_8*C_4</math>]] || No || 3(?) || No || || [[References|[Sa13b] ]] || Block invariants known<br />
|-<br />
|16 || [[(C2)^4|14]] || [[(C2)^4|<math>(C_2)^4</math>]] || <math>\mathcal{O}</math> || 16(16) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Ea18] ]] ||<br />
|}<br />
<br />
The table for defect groups of order 32 takes as its starting point Table 13.1 of Sambale's book [[References|[Sa14]]].<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>|D|=32</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Donovan (w.r.t)?<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|32 || [[C32|1]] || [[C32|<math>C_{32}</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|32 || [[MNA(2,2)|2]] || [[MNA(2,2)|<math>MNA(2,2)</math>]] || <math>\mathcal{O}</math> || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EKS12]]] ||<br />
|-<br />
|32 || [[C8xC4|3]] || [[C8xC4|<math>C_8 \times C_4</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|-<br />
|32 || [[C8:C4|4]] || [[C8:C4|<math>C_8:C_4</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[MNA(3,1)|5]] || [[MNA(3,1)|<math>MNA(3,1)</math>]] || No || || || || [[References#S|[Sa11] ]] || Invariants known<br />
|-<br />
|32 || [[MNA(2,1):C2|6]] || [[MNA(3,1):C2|<math>MNA(2,1):C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,7)|7]] || [[SmallGroup(32,7)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] || <math>M_{16}:C_2</math><br />
|-<br />
|32 || [[2.MNA(2,1)|8]] || [[2.MNA(2,1)|<math>2.MNA(2,1)</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[D8:C4|9]] || [[D8:C4|<math>D_8:C_4</math>]] || No || || || || [[References#S|[Sa14,10.23]]] || Invariants known<br />
|-<br />
|32 || [[Q8:C4|10]] || [[Q8:C4|<math>Q_8:C_4</math>]] || No || || || || [[References#S|[Sa14,10.25]]] || Invariants known<br />
|-<br />
|32 || [[C4wrC2|11]] || [[C4wrC2|<math>C_4 \wr C_2</math>]] || No || 6(6) || No || || [[References#K|[Ku80]]], [[References#K|[KoLaSa23]]] || Invariants known. Principal blocks classified up to source algebra equivalence in [[References#K|[KoLaSa23]]]<br />
|-<br />
|32 || [[C4:C8|12]] || [[C4:C8|<math>C_4:C_8</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[C8:C4a|13]] || [[C8:C4a|<math>C_8:C_4=\langle a,b|a^8=b^4=1, ba=a^3b \rangle</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[C8:C4b|14]] || [[C8:C4b|<math>C_8:C_4=\langle a,b|a^8=b^4=1, ba=a^7b \rangle</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,15)|15]] || [[SmallGroup(32,15)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[C16xC2|16]] || [[C16xC2|<math>C_{16} \times C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|32 || [[M32|17]] || [[M32|<math>M_{32}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[D32|18]] || [[D32|<math>D_{32}</math>]] || <math>k</math> || 5(?) || <math>k</math> || <math>k</math> || [[References#E|[Er87]]] || Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] <br />
|-<br />
|32 || [[SD32|19]] || [[SD32|<math>SD_{32}</math>]] || <math>k</math> || || || || ||<br />
|-<br />
|32 || [[Q32|20]] || [[Q32|<math>Q_{32}</math>]] || No || || || || [[References#E|[Er88a], [Er88b], [Ho97]]] || Two possibly infinite families when <math>l(B)=2</math>. Classified over <math>\mathcal{O}</math> when <math>l(B)=3</math> in [[References#E|[Ei16]]]. Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20b]]]<br />
|-<br />
|32 || [[C4xC4xC2|21]] || [[C4xC4xC2|<math>C_4 \times C_4 \times C_2</math>]] || <math>\mathcal{O}</math> || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EKKS14]]] <br />
|-<br />
|32 || [[MNA(2,1)xC2|22]] || [[MNA(2,1)xC2|<math>MNA(2,1) \times C_2</math>]] || No || || || || [[References#S|[Sa14,10.25]]] || Invariants known<br />
|-<br />
|32 || [[(C4:C4)xC2|23]] || [[(C4:C4)xC2|<math>(C_4:C_4) \times C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,24)|24]] || [[SmallGroup(32,24)]]<!--<math>(C_4 \times C_4):C_2=\langle a,b,c \mid a^4 = b^4 = c^2 = e, ab = ba, ac = ca, cb = a^2bc \rangle</math>]]--> || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[D8xC4|25]] || [[D8xC4|<math>D_8 \times C_4</math>]] || No || || || || [[References#S|[Sa14,9.7]]] ||<br />
Invariants known<br />
|-<br />
|32 || [[Q8xC4|26]] || [[Q8xC4|<math>Q_8 \times C_4</math>]] || <math>\mathcal{O}</math> || 3(3) || No || || [[References#E|[EL20]]] || Invariants known by [[References#S|[Sa14,9.28]]]<br />
|-<br />
|32 || [[SmallGroup(32,27)|27]] || [[SmallGroup(32,27)]]<!--|<math>(C_4 \times C_4):C_2=\langle x,y,z,a,b \mid x^2 = y^2 = z^2 = a^2 = b^2 = e, xy = yx, xz, = zx, yz = zy, aza^{-1} = xz, bzb^{-1} = yz, ax = xa, ay = ya, bx = xb, by = yb \rangle</math>]]--> || No || || || || ||<br />
|-<br />
|32 || [[SmallGroup(32,28)|28]] || [[SmallGroup(32,28)]] || No || || || || [[References#S|[Sa14,13.11]]] || Invariants known<br />
|-<br />
|32 || [[SmallGroup(32,29)|29]] || [[SmallGroup(32,29)]] || No || || || || [[References#S|[Sa14,13.11]]] || Invariants known<br />
|-<br />
|32 || [[SmallGroup(32,30)|30]] || [[SmallGroup(32,30)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,31)|31]] || [[SmallGroup(32,31)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,32)|32]] || [[SmallGroup(32,32)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,33)|33]] || [[SmallGroup(32,33)]] || No || || || || [[References|[Sa14,13.12]]] || Invariants partly known<br />
|-<br />
|32 || [[SmallGroup(32,34)|34]] || [[SmallGroup(32,34)]] || No || || || || [[References|[Sa14,13.12]]] || Invariants partly known<br />
|-<br />
|32 || [[C4:Q8|35]] || [[C4:Q8|<math>C_4:Q_8</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[C8xC2xC2|36]] || [[C8xC2xC2|<math>C_8 \times C_2 \times C_2</math>]] || <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EL18a]]] ||<br />
|-<br />
|32 || [[M16xC2|37]] || [[M16xC2|<math>M_{16} \times C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[D8*C8|38]] || [[D8*C8|<math>D_8*C_8</math>]] || No || || || || [[References#S|[Sa14,9.18]]] || Invariants known<br />
|-<br />
|32 || [[D16xC2|39]] || [[D16xC2|<math>D_{16} \times C_2</math>]] || No || || || || [[References#S|[Sa14,9.7]]] || Invariants known<br />
|-<br />
|32 || [[SD16xC2|40]] || [[SD16xC2|<math>SD_{16} \times C_2</math>]] || No || || || || [[References#S|[Sa14,9.37]]] || Invariants known<br />
|-<br />
|32 || [[Q16xC2|41]] || [[Q16xC2|<math>Q_{16} \times C_2</math>]] || No || || || || [[References#S|[Sa14,9.28]]] || Invariants known<br />
|-<br />
|32 || [[D16*C4|42]] || [[D16*C4|<math>D_{16}*C_4</math>]] || No || || || || [[References|[Sa14,9.18]]] || Invariants known<br />
|-<br />
|32 || [[SmallGroup(32,43)|43]] || [[SmallGroup(32,43)]] || No || || || || ||<br />
|-<br />
|32 || [[SmallGroup(32,44)|44]] || [[SmallGroup(32,44)]] || No || || || || ||<br />
|-<br />
|32 || [[C4xC2xC2xC2|45]] || [[C4xC2xC2xC2|<math>C_4 \times C_2 \times C_2 \times C_2</math>]] || <math>\mathcal{O}</math> || 8(8) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EL23]]] ||<br />
|-<br />
|32 || [[D8xC2xC2|46]] || [[D8xC2xC2|<math>D_8 \times C_2 \times C_2</math>]] || No || || || || ||<br />
|-<br />
|32 || [[Q8xC2xC2|47]] || [[Q8xC2xC2|<math>Q_8 \times C_2 \times C_2</math>]] || No || || || || ||<br />
|-<br />
|32 || [[D8*C4xC2|48]] || [[D8*C4xC2|<math>(D_8*C_4) \times C_2</math>]] || No || || || || ||<br />
|-<br />
|32 || [[D8*D8|49]] || [[D8*D8|<math>D_8*D_8</math>]] || No || || || || [[References#S|[Sa13c]]] || Invariants partly known<br />
|-<br />
|32 || [[D8*Q8|50]] || [[D8*Q8|<math>D_8*Q_8</math>]] || No || || || || [[References#S|[Sa13c]]] || Invariants partly known<br />
|-<br />
|32 || [[(C2)^5|51]] || [[(C2)^5|<math>(C_2)^5</math>]] || <math>\mathcal{O}</math> || 34 (34) || <math>\mathcal{O}</math> || || [[References#A|[Ar19]]] || Derived eq. classes determined for 30 of the 34 Morita eq. classes. <br />
|}<br />
<br />
<!--<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>|D|=64</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Donovan (w.r.t)?<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|64 || [[C64|1]] || [[C64|<math>C_{64}</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
|64 || [[C8xC8|2]] || [[C8xC8|<math>C_8 \times C_8</math>]] || <math>\mathcal{O}</math> ||2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EKKS14]]]||<br />
|-<br />
|64 || [[SmallGroup(64,3)|3]] || [[SmallGroup(64,3)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_8:C_8</math><br />
|-<br />
|64 || [[(C2xC2xC2):C8|4]] || [[(C2xC2xC2):C8|<math>(C_2)^3:C_8</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,5)|5]] || [[SmallGroup(64,5)]] || No || || || || ||<br />
|-<br />
|64 || [[(D8:C8|6]] || [[D8:C8|<math>D_8:C_8</math>]] || No || || || || ||<br />
|-<br />
|64 || [[(Q8:C8|7]] || [[Q8:C8|<math>Q_8:C_8</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,8)|8]] || [[SmallGroup(64,8)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,9)|9]] || [[SmallGroup(64,9)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,10)|10]] || [[SmallGroup(64,10)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,11)|11]] || [[SmallGroup(64,11)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,12)|12]] || [[SmallGroup(64,12)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,13)|13]] || [[SmallGroup(64,13)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,14)|14]] || [[SmallGroup(64,14)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,15)|15]] || [[SmallGroup(64,15)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_8:C_8</math><br />
|-<br />
|64 || [[SmallGroup(64,16)|16]] || [[SmallGroup(64,16)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_8:C_8</math><br />
|-<br />
|64 || [[SmallGroup(64,17)|17]] || [[SmallGroup(64,17)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,18)|18]] || [[SmallGroup(64,18)]] || No || || || || || <math>(C_4 \times C_4):C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,19)|19]] || [[SmallGroup(64,19)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,20)|20]] || [[SmallGroup(64,20)]] || No || || || || || <math>(C_4 \times C_4):C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,21)|21]] || [[SmallGroup(64,21)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,22)|22]] || [[SmallGroup(64,22)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,23)|23]] || [[SmallGroup(64,23)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,24)|24]] || [[SmallGroup(64,24)]] || No || || || || || <math>M_{16}:C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,25)|25]] || [[SmallGroup(64,25)]] || No || || || || || <math>M_{16}:C_4</math><br />
|-<br />
|64 || [[C16xC4|26]] || [[C16xC4|<math>C_{16} \times C_4</math>]]|| <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|64 || [[SmallGroup(64,27)|27]] || [[SmallGroup(64,27)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_{16}:C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,28)|28]] || [[SmallGroup(64,28)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_{16}:C_4</math><br />
|-<br />
|64 || [[(C2xC2):C16|29]] || [[(C2xC2):C16|<math>(C_2)^2:C_{16}</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,30)|30]] || [[SmallGroup(64,30)]] || No || || || || || <math>M_{32}:C_2</math><br />
|-<br />
|64 || [[SmallGroup(64,31)|31]] || [[SmallGroup(64,31)]] || No || || || || || <math>M_{32}:C_2</math><br />
|-<br />
|64 || [[C2wrC4|32]] || [[(C2wrC4|<math>C_2 \wr C_4</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,33)|33]] || [[SmallGroup(64,33)]] || No || || || || || <br />
|-<br />
|64 || [[SmallGroup(64,34)|34]] || [[SmallGroup(64,31)]] || No || || || || || <math>(C_4 \times C_4):C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,35)|35]] || [[SmallGroup(64,35)]] || No || || || || || <math>(C_4 \times C_4):C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,36)|36]] || [[SmallGroup(64,36)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,37)|37]] || [[SmallGroup(64,37)]] || No || || || || || <math>C_4:Q_8</math>, fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,38)|38]] || [[SmallGroup(64,38)]] || No || || || || || <math>D_{16}:C_4</math>, fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,39)|39]] || [[SmallGroup(64,39)]] || No || || || || || <math>Q_{16}:C_2</math><br />
|-<br />
|64 || [[SmallGroup(64,40)|40]] || [[SmallGroup(64,40)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,41)|41]] || [[SmallGroup(64,41)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,42)|42]] || [[SmallGroup(64,42)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,43)|43]] || [[SmallGroup(64,43)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[C4:C16|44]] || [[C4:C16|<math>C_4:C_{16}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_4:C_{16}</math><br />
|-<br />
|64 || [[SmallGroup(64,45)|45]] || [[SmallGroup(64,45)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,46)|46]] || [[SmallGroup(64,46)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,47)|47]] || [[SmallGroup(64,47)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_{16}:C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,48)|48]] || [[SmallGroup(64,48)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_{16}:C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,48)|49]] || [[SmallGroup(64,49)]] || No || || || || ||<br />
|- <br />
|64 || [[C32xC2|50]] || [[C32xC2|<math>C_{32} \times C_2</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|64 || [[M6(2)|51]] || [[M6(2)|<math>M_6(2)</math>]] || No || || || || ||<br />
|-<br />
|64 || [[D64|52]] || [[D64|<math>D_{64}</math>]] || No || <math>k</math> || || || ||<br />
|-<br />
|64 || [[SD64|53]] || [[SD64|<math>SD_{64}</math>]] || No || <math>k</math> || || || ||<br />
|-<br />
|64 || [[Q64|54]] || [[Q64|<math>Q_{64}</math>]] || No || || || || ||<br />
|- <br />
|64 || [[C4xC4xC4|55]] || [[C4xC4xC4|<math>C_4 \times C_4 \times C_4</math>]] || <math>\mathcal{O}</math> ||4(4) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EL18a]]]||<br />
|-<br />
|64 || [[SmallGroup(64,56)|56]] || [[SmallGroup(64,56)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,57)|57]] || [[SmallGroup(64,57)]] || No || || || || ||<br />
|- <br />
|64 || [[C4x(C2xC2):C4|58]] || [[C4x(C2xC2):C4|<math>C_{4} \times (C_2 \times C_2):C_4</math>]] || || || || || || Fusion trivial?<br />
|- <br />
|64 || [[C4x(C4:C4)|59]] || [[C4x(C4:C4)|<math>C_{4} \times (C_4:C_4)</math>]] || || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,60)|60]] || [[SmallGroup(64,60)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,61)|61]] || [[SmallGroup(64,61)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,62)|62]] || [[SmallGroup(64,62)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,63)|63]] || [[SmallGroup(64,63)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|-<br />
|64 || [[SmallGroup(64,64)|64]] || [[SmallGroup(64,64)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,65)|65]] || [[SmallGroup(64,65)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,66)|66]] || [[SmallGroup(64,66)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,67)|67]] || [[SmallGroup(64,67)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,68)|68]] || [[SmallGroup(64,68)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|-<br />
|64 || [[SmallGroup(64,69)|69]] || [[SmallGroup(64,69)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,70)|70]] || [[SmallGroup(64,70)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|-<br />
|64 || [[SmallGroup(64,71)|71]] || [[SmallGroup(64,71)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,72)|72]] || [[SmallGroup(64,72)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|-<br />
|64 || [[SmallGroup(64,73)|73]] || [[SmallGroup(64,73)]] || No || || || || ||<br />
|-<br />
|64 || [[(C2)^3:Q8|74]] || [[(C2)^3:Q8|<math>(C_2)^3:Q_8</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,75)|75]] || [[SmallGroup(64,75)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,76)|76]] || [[SmallGroup(64,76)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|-<br />
|64 || [[SmallGroup(64,77)|77]] || [[SmallGroup(64,77)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,78)|78]] || [[SmallGroup(64,78)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,79)|79]] || [[SmallGroup(64,79)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|-<br />
|64 || [[SmallGroup(64,80)|80]] || [[SmallGroup(64,80)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,81)|81]] || [[SmallGroup(64,81)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|-<br />
|64 || [[SmallGroup(64,82)|82]] || [[SmallGroup(64,82)]] || <math>\mathcal{O}</math> || 6(6) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[Ea24]]] || Sylow 2-subgroup of <math>Sz(8)</math><br />
|-<br />
|64 || [[C8xC4xC2|83]] || [[C8xC4xC2|<math>C_{8} \times C_4 \times C_2</math>]]|| <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|64 || [[C2x(C8:C4)|84]] || [[C2x(C8:C4)|<math>C_{2} \times (C_8:C_4)</math>]]|| No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[M4(2)xC4|85]] || [[M4(2)xC4|<math>M_4(2) \times C_4</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,86)|86]] || [[SmallGroup(64,86)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[C2x(C2xC2):C8|87]] || [[C2x(C2xC2):C8|<math>C_{2} \times (C_2 \times C_2):C_8</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,88)|88]] || [[SmallGroup(64,88)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[(C8xC2xC2):C2|89]] || [[(C8xC2xC2):C2|<math>(C_8 \times C_2 \times C_2):C_2</math>]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[C2x(C2xC2xC2):C4|90]] || [[C2x(C2xC2xC2):C4|<math>C_2 \times (C_2 \times C_2 \times C_2):C_4</math>]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,91)|91]] || [[SmallGroup(64,91)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,92)|92]] || [[SmallGroup(64,92)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,93)|93]] || [[SmallGroup(64,93)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,94)|94]] || [[SmallGroup(64,94)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[C2x(D_8:C4)|95]] || [[C2x(D_8:C4)|<math>C_{2} \times (D_8:C_4)</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[C2x(Q_8:C4)|96]] || [[C2x(Q_8:C4)|<math>C_{2} \times (Q_8:C_4)</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,97)|97]] || [[SmallGroup(64,97)]] || No || || || || || <br />
|-<br />
|64 || [[SmallGroup(64,98)|98]] || [[SmallGroup(64,98)]] || No || || || || || <br />
|-<br />
|64 || [[SmallGroup(64,99)|99]] || [[SmallGroup(64,99)]] || No || || || || || <br />
|-<br />
|64 || [[SmallGroup(64,100)|100]] || [[SmallGroup(64,100)]] || No || || || || || <br />
|-<br />
|64 || [[C2x(C4wrC2)|101]] || [[C2x(C4wrC2)|<math>C_{2} \times (C_4 \wr C_2)</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[(C4xC4)(C2:C2)|102]] || [[(C4xC4)(C2:C2)|<math>(C_4 \times C_4):(C_2 \times C_2)</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[C2x(C4:C8)|103]] || [[C2x(C4:C8)|<math>C_{2} \times (C_4:C_8)</math>]]|| No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[C4:M4(2)|104]] || [[C4:M4(2)|<math>C_{4}:M_4(2)</math>]]|| No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,105)|105]] || [[SmallGroup(64,105)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,106)|106]] || [[SmallGroup(64,106)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,107)|107]] || [[SmallGroup(64,107)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,108)|108]] || [[SmallGroup(64,108)]] || No || || || || ||<br />
|-<br />
|64 || [[M4(2):C4|109]] || [[M4(2):C4|<math>M_4(2):C_4</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,110)|110]] || [[SmallGroup(64,110)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,111)|111]] || [[SmallGroup(64,111)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,112)|112]] || [[SmallGroup(64,112)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|-<br />
|64 || [[SmallGroup(64,113)|113]] || [[SmallGroup(64,113)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,114)|114]] || [[SmallGroup(64,114)]] || No || || || || ||<br />
|-<br />
|64 || [[D8xC8|115]] || [[D8xC8|<math>(D_8 \times C_8</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,116)|116]] || [[SmallGroup(64,116)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,117)|117]] || [[SmallGroup(64,117)]] || No || || || || ||<br />
|-<br />
|64 || [[D16xC4|118]] || [[D16xC4|<math>(D_{16} \times C_4</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[SD16xC4|119]] || [[SD16xC4|<math>(SD_{16} \times C_4</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[Q16xC4|120]] || [[Q16xC4|<math>(Q_{16} \times C_4</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[SD16:C4|121]] || [[SD16:C4|<math>(SD_{16}:C_4</math>]]|| No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[Q16:C4|122]] || [[Q16:C4|<math>(Q_{16}:C_4</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[D16:C4|123]] || [[D16:C4|<math>(D_{16}:C_4</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,124)|124]] || [[SmallGroup(64,124)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,125)|125]] || [[SmallGroup(64,125)]] || No || || || || ||<br />
|-<br />
|64 || [[Q8xC8|126]] || [[Q8xC8|<math>(Q_{8} \times C_8</math>]]|| <math>\mathcal{O}</math> || 3(3) || || || [[References#E|[EL20]]] || Invariants known by [[References#S|[Sa14,9.28]]]<br />
|-<br />
|64 || [[SmallGroup(64,127)|127]] || [[SmallGroup(64,127)]] || No || || || || ||<br />
|-<br />
|64 || [[(C2xC2):D16|128]] || [[(C2xC2)D16|<math>(C_2 \times C_2):D_{16}</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[Q8:D8|129]] || [[Q8:D8|<math>Q_8:D_{8}</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[D8:D8|130]] || [[D8:D8|<math>D_8:D_{8}</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,245)|245]] || [[SmallGroup(64,245)]] || <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[Ea24]]] || Sylow 2-subgroup of <math>PSU_3(4)</math><br />
<br />
|}--><br />
<br />
==Blocks for <math>p=3</math>==<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>1 \leq |D| \leq 27</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|- <br />
| 3 || [[C3|1]] || [[C3|<math>C_3</math>]] || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
|9 || [[C9|1]] ||[[C9|<math>C_9</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|-<br />
|9 || [[C3xC3|2]] || [[C3xC3|<math>C_3 \times C_3</math>]] || || || || ||<br />
|-<br />
|27 || [[C27|1]] || [[C27|<math>C_{27}</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|27 || [[C9xC3|2]] || [[C9xC3|<math>C_9 \times C_3</math>]] || || || || ||<br />
|-<br />
|27 || [[3_+^3|3]] || [[3_+^3|<math>3_+^{1+2}</math>]] || || || || ||<br />
|-<br />
|27 || [[3_-^3|4]] || [[3_-^3|<math>3_-^{1+2}</math>]] || || || || ||<br />
|-<br />
|27 || [[C3xC3xC3|5]] || [[C3xC3xC3|<math>C_3 \times C_3 \times C_3</math>]] || || || || ||<br />
|}<br />
<br />
==Blocks for <math>p=5</math>==<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>5 \leq |D| \leq 125</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|- <br />
|5 || [[C5|1]] || [[C5|<math>C_5</math>]] ||6(6) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
|25 || [[C25|1]] ||[[C25|<math>C_{25}</math>]] || 6(6) || No || <math>\mathcal{O}</math> || || Max 12 classes <br />
|-<br />
|25 || [[C5xC5|2]] || [[C5xC5|<math>C_5 \times C_5</math>]] || || || || ||<br />
|- <br />
|125 || [[C125|1]] ||[[C125|<math>C_{125}</math>]] || || || || || <br />
|-<br />
|125 || [[C25xC5|2]] || [[C25xC5|<math>C_{25} \times C_5</math>]] || || || || || <br />
|-<br />
|125 || [[5_+^3|3]] || [[5_+^3|<math>5_+^{1+2}</math>]] || 62(62) || <math>\mathcal{O}</math> || || [[References#A|[AE23]]] || Inertial quotients are consistent within classes<br />
|-<br />
|125 || [[5_-^3|4]] || [[5_-^3|<math>5_-^{1+2}</math>]] || || || || ||<br />
|-<br />
|125 || [[C5xC5xC5|5]] || [[C5xC5xC5|<math>C_5 \times C_5 \times C_5</math>]] || || || || ||<br />
|}<br />
<br />
==Blocks for <math>p\geq 7</math>==<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>|D|</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|- <br />
|7 || [[C7|1]] || [[C7|<math>C_7</math>]] ||14(14) ||No || <math>\mathcal{O}</math> || ||Max 21 classes <br />
|- <br />
|11|| [[C11|1]] || [[C11|<math>C_{11}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|- <br />
|13 || [[C13|1]] || [[C13|<math>C_{13}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|-<br />
|17|| [[C17|1]] || [[C17|<math>C_{17}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|- <br />
|19 || [[C19|1]] || [[C19|<math>C_{19}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|-<br />
|23 || [[C23|1]] || [[C23|<math>C_{23}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|}</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Classification_by_p-group&diff=1238
Classification by p-group
2024-01-09T18:49:39Z
<p>Charles Eaton: </p>
<hr />
<div>'''Classification of Morita equivalences for blocks with a given defect group'''<br />
<br />
On this page we list classifications of Morita equivalence classes for each isomorphism class of p-groups in turn. [[Generic classifications by p-group class|Generic classifications for classes of p-groups can be found here]].<br />
<br />
See [[Labelling for Morita equivalence classes|this page]] for a description of the labelling conventions.<br />
<br />
== Blocks for <math> p=2 </math> ==<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>1 \leq |D| \leq 8</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|- <br />
| 2 || [[C2|1]] || [[C2|<math>C_2</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
| 4 || [[C4|1]] || [[C4|<math>C_4</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
| 4 || [[C2xC2|2]] || [[C2xC2|<math>C_2 \times C_2</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Er82], [Li94] ]] ||<br />
|- <br />
|8 || [[C8|1]] || [[C8|<math>C_8</math>]] ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|8 || [[C4xC2|2]] || [[C4xC2|<math>C_4 \times C_2</math>]] ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|8 || [[D8|3]] || [[D8|<math>D_8</math>]] ||6(?) || <math>k</math> || <math>k</math> || [[References|[Er87] ]] || Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] <br />
|-<br />
|8 || [[Q8|4]] || [[Q8|<math>Q_8</math>]] ||3(3) || <math>\mathcal{O}</math> || <math>k</math> || [[References|[Er88a], [Er88b], [HKL07], [Ei16]]] || <br />
|-<br />
|8 || [[C2xC2xC2|5]] || [[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]] || 8(8) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References| [Ea16]]] || Uses CFSG<br />
|}<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>|D|=16</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Donovan (w.r.t)?<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|16 || [[C16|1]] || [[C16|<math>C_{16}</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|16 || [[C4xC4|2]] || [[C4xC4|<math>C_4 \times C_4</math>]] || <math>\mathcal{O}</math> || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[EKKS14] ]] ||<br />
|-<br />
|16 || [[MNA(2,1)|3]] || [[MNA(2,1)]] || No || 3(?) || No || || [[References|[Sa11] ]] || Block invariants known<br />
|-<br />
|16 || [[C4:C4|4]] || [[C4:C4|<math>C_4:C_4</math>]] || <math>\mathcal{O}</math>|| 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[CG12], [Sa12b]]] ||<br />
|-<br />
|16 || [[C8xC2|5]] || [[C8xC2|<math>C_8 \times C_2</math>]]|| <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|16 || [[M16|6]] || [[M16|<math>M_{16}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[CG12], [Sa12b] ]] || <br />
|-<br />
|16 || [[D16|7]] || [[D16|<math>D_{16}</math>]] || <math>k</math>|| 5(?) || <math>k</math> || <math>k</math> || [[References|[Er87]]] || Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] <br />
|-<br />
|16 || [[SD16|8]] || [[SD16|<math>SD_{16}</math>]] || <math>k</math> || 7(?) || || || [[References|[Er88c], [Er90b]]] || Two other possible classes<br />
|-<br />
|16 || [[Q16|9]] || [[Q16|<math>Q_{16}</math>]] || No || 6(?) || || <math>k</math> || [[References|[Er88a], [Er88b], [Ho97]]] || Two possibly infinite families when <math>l(B)=2</math>. Classified over <math>\mathcal{O}</math> when <math>l(B)=3</math> in [[References#E|[Ei16]]]. Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20b]]]<br />
|-<br />
|16 || [[C4xC2xC2|10]] || [[C4xC2xC2|<math>C_4 \times C_2 \times C_2</math>]]|| <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[EL18a]]] ||<br />
|-<br />
|16 || [[D8xC2|11]] || [[D8xC2|<math>D_8 \times C_2</math>]] || No || || || || [[References|[Sa12] ]] || Block invariants known<br />
|-<br />
|16 || [[Q8xC2|12]] || [[Q8xC2|<math>Q_8 \times C_2</math>]] || <math>\mathcal{O}</math> || 3(3) || No || || [[References#E|[EL20]]] || Block invariants known by [[References#S|[Sa13]]]<br />
|-<br />
|16 || [[D8*C4|13]] || [[D8*C4|<math>D_8*C_4</math>]] || No || 3(?) || No || || [[References|[Sa13b] ]] || Block invariants known<br />
|-<br />
|16 || [[(C2)^4|14]] || [[(C2)^4|<math>(C_2)^4</math>]] || <math>\mathcal{O}</math> || 16(16) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Ea18] ]] ||<br />
|}<br />
<br />
The table for defect groups of order 32 takes as its starting point Table 13.1 of Sambale's book [[References|[Sa14]]].<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>|D|=32</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Donovan (w.r.t)?<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|32 || [[C32|1]] || [[C32|<math>C_{32}</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|32 || [[MNA(2,2)|2]] || [[MNA(2,2)|<math>MNA(2,2)</math>]] || <math>\mathcal{O}</math> || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EKS12]]] ||<br />
|-<br />
|32 || [[C8xC4|3]] || [[C8xC4|<math>C_8 \times C_4</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|-<br />
|32 || [[C8:C4|4]] || [[C8:C4|<math>C_8:C_4</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[MNA(3,1)|5]] || [[MNA(3,1)|<math>MNA(3,1)</math>]] || No || || || || [[References#S|[Sa11] ]] || Invariants known<br />
|-<br />
|32 || [[MNA(2,1):C2|6]] || [[MNA(3,1):C2|<math>MNA(2,1):C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,7)|7]] || [[SmallGroup(32,7)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] || <math>M_{16}:C_2</math><br />
|-<br />
|32 || [[2.MNA(2,1)|8]] || [[2.MNA(2,1)|<math>2.MNA(2,1)</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[D8:C4|9]] || [[D8:C4|<math>D_8:C_4</math>]] || No || || || || [[References#S|[Sa14,10.23]]] || Invariants known<br />
|-<br />
|32 || [[Q8:C4|10]] || [[Q8:C4|<math>Q_8:C_4</math>]] || No || || || || [[References#S|[Sa14,10.25]]] || Invariants known<br />
|-<br />
|32 || [[C4wrC2|11]] || [[C4wrC2|<math>C_4 \wr C_2</math>]] || No || 6(6) || No || || [[References#K|[Ku80]]], [[References#K|[KoLaSa23]]] || Invariants known. Principal blocks classified up to source algebra equivalence in [[References#K|[KoLaSa23]]]<br />
|-<br />
|32 || [[C4:C8|12]] || [[C4:C8|<math>C_4:C_8</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[C8:C4a|13]] || [[C8:C4a|<math>C_8:C_4=\langle a,b|a^8=b^4=1, ba=a^3b \rangle</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[C8:C4b|14]] || [[C8:C4b|<math>C_8:C_4=\langle a,b|a^8=b^4=1, ba=a^7b \rangle</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,15)|15]] || [[SmallGroup(32,15)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[C16xC2|16]] || [[C16xC2|<math>C_{16} \times C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|32 || [[M32|17]] || [[M32|<math>M_{32}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[D32|18]] || [[D32|<math>D_{32}</math>]] || <math>k</math> || 5(?) || <math>k</math> || <math>k</math> || [[References#E|[Er87]]] || Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] <br />
|-<br />
|32 || [[SD32|19]] || [[SD32|<math>SD_{32}</math>]] || <math>k</math> || || || || ||<br />
|-<br />
|32 || [[Q32|20]] || [[Q32|<math>Q_{32}</math>]] || No || || || || [[References#E|[Er88a], [Er88b], [Ho97]]] || Two possibly infinite families when <math>l(B)=2</math>. Classified over <math>\mathcal{O}</math> when <math>l(B)=3</math> in [[References#E|[Ei16]]]. Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20b]]]<br />
|-<br />
|32 || [[C4xC4xC2|21]] || [[C4xC4xC2|<math>C_4 \times C_4 \times C_2</math>]] || <math>\mathcal{O}</math> || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EKKS14]]] <br />
|-<br />
|32 || [[MNA(2,1)xC2|22]] || [[MNA(2,1)xC2|<math>MNA(2,1) \times C_2</math>]] || No || || || || [[References#S|[Sa14,10.25]]] || Invariants known<br />
|-<br />
|32 || [[(C4:C4)xC2|23]] || [[(C4:C4)xC2|<math>(C_4:C_4) \times C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,24)|24]] || [[SmallGroup(32,24)]]<!--<math>(C_4 \times C_4):C_2=\langle a,b,c \mid a^4 = b^4 = c^2 = e, ab = ba, ac = ca, cb = a^2bc \rangle</math>]]--> || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[D8xC4|25]] || [[D8xC4|<math>D_8 \times C_4</math>]] || No || || || || [[References#S|[Sa14,9.7]]] ||<br />
Invariants known<br />
|-<br />
|32 || [[Q8xC4|26]] || [[Q8xC4|<math>Q_8 \times C_4</math>]] || <math>\mathcal{O}</math> || 3(3) || No || || [[References#E|[EL20]]] || Invariants known by [[References#S|[Sa14,9.28]]]<br />
|-<br />
|32 || [[SmallGroup(32,27)|27]] || [[SmallGroup(32,27)]]<!--|<math>(C_4 \times C_4):C_2=\langle x,y,z,a,b \mid x^2 = y^2 = z^2 = a^2 = b^2 = e, xy = yx, xz, = zx, yz = zy, aza^{-1} = xz, bzb^{-1} = yz, ax = xa, ay = ya, bx = xb, by = yb \rangle</math>]]--> || No || || || || ||<br />
|-<br />
|32 || [[SmallGroup(32,28)|28]] || [[SmallGroup(32,28)]] || No || || || || [[References#S|[Sa14,13.11]]] || Invariants known<br />
|-<br />
|32 || [[SmallGroup(32,29)|29]] || [[SmallGroup(32,29)]] || No || || || || [[References#S|[Sa14,13.11]]] || Invariants known<br />
|-<br />
|32 || [[SmallGroup(32,30)|30]] || [[SmallGroup(32,30)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,31)|31]] || [[SmallGroup(32,31)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,32)|32]] || [[SmallGroup(32,32)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,33)|33]] || [[SmallGroup(32,33)]] || No || || || || [[References|[Sa14,13.12]]] || Invariants partly known<br />
|-<br />
|32 || [[SmallGroup(32,34)|34]] || [[SmallGroup(32,34)]] || No || || || || [[References|[Sa14,13.12]]] || Invariants partly known<br />
|-<br />
|32 || [[C4:Q8|35]] || [[C4:Q8|<math>C_4:Q_8</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[C8xC2xC2|36]] || [[C8xC2xC2|<math>C_8 \times C_2 \times C_2</math>]] || <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EL18a]]] ||<br />
|-<br />
|32 || [[M16xC2|37]] || [[M16xC2|<math>M_{16} \times C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[D8*C8|38]] || [[D8*C8|<math>D_8*C_8</math>]] || No || || || || [[References#S|[Sa14,9.18]]] || Invariants known<br />
|-<br />
|32 || [[D16xC2|39]] || [[D16xC2|<math>D_{16} \times C_2</math>]] || No || || || || [[References#S|[Sa14,9.7]]] || Invariants known<br />
|-<br />
|32 || [[SD16xC2|40]] || [[SD16xC2|<math>SD_{16} \times C_2</math>]] || No || || || || [[References#S|[Sa14,9.37]]] || Invariants known<br />
|-<br />
|32 || [[Q16xC2|41]] || [[Q16xC2|<math>Q_{16} \times C_2</math>]] || No || || || || [[References#S|[Sa14,9.28]]] || Invariants known<br />
|-<br />
|32 || [[D16*C4|42]] || [[D16*C4|<math>D_{16}*C_4</math>]] || No || || || || [[References|[Sa14,9.18]]] || Invariants known<br />
|-<br />
|32 || [[SmallGroup(32,43)|43]] || [[SmallGroup(32,43)]] || No || || || || ||<br />
|-<br />
|32 || [[SmallGroup(32,44)|44]] || [[SmallGroup(32,44)]] || No || || || || ||<br />
|-<br />
|32 || [[C4xC2xC2xC2|45]] || [[C4xC2xC2xC2|<math>C_4 \times C_2 \times C_2 \times C_2</math>]] || <math>\mathcal{O}</math> || 8(8) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EL23]]] ||<br />
|-<br />
|32 || [[D8xC2xC2|46]] || [[D8xC2xC2|<math>D_8 \times C_2 \times C_2</math>]] || No || || || || ||<br />
|-<br />
|32 || [[Q8xC2xC2|47]] || [[Q8xC2xC2|<math>Q_8 \times C_2 \times C_2</math>]] || No || || || || ||<br />
|-<br />
|32 || [[D8*C4xC2|48]] || [[D8*C4xC2|<math>(D_8*C_4) \times C_2</math>]] || No || || || || ||<br />
|-<br />
|32 || [[D8*D8|49]] || [[D8*D8|<math>D_8*D_8</math>]] || No || || || || [[References#S|[Sa13c]]] || Invariants partly known<br />
|-<br />
|32 || [[D8*Q8|50]] || [[D8*Q8|<math>D_8*Q_8</math>]] || No || || || || [[References#S|[Sa13c]]] || Invariants partly known<br />
|-<br />
|32 || [[(C2)^5|51]] || [[(C2)^5|<math>(C_2)^5</math>]] || <math>\mathcal{O}</math> || 34 (34) || <math>\mathcal{O}</math> || || [[References#A|[Ar19]]] || Derived eq. classes determined for 30 of the 34 Morita eq. classes. <br />
|}<br />
<br />
<!--<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>|D|=64</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Donovan (w.r.t)?<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|64 || [[C64|1]] || [[C64|<math>C_{64}</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
|64 || [[C8xC8|2]] || [[C8xC8|<math>C_8 \times C_8</math>]] || <math>\mathcal{O}</math> ||2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EKKS14]]]||<br />
|-<br />
|64 || [[SmallGroup(64,3)|3]] || [[SmallGroup(64,3)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_8:C_8</math><br />
|-<br />
|64 || [[(C2xC2xC2):C8|4]] || [[(C2xC2xC2):C8|<math>(C_2)^3:C_8</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,5)|5]] || [[SmallGroup(64,5)]] || No || || || || ||<br />
|-<br />
|64 || [[(D8:C8|6]] || [[D8:C8|<math>D_8:C_8</math>]] || No || || || || ||<br />
|-<br />
|64 || [[(Q8:C8|7]] || [[Q8:C8|<math>Q_8:C_8</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,8)|8]] || [[SmallGroup(64,8)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,9)|9]] || [[SmallGroup(64,9)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,10)|10]] || [[SmallGroup(64,10)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,11)|11]] || [[SmallGroup(64,11)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,12)|12]] || [[SmallGroup(64,12)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,13)|13]] || [[SmallGroup(64,13)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,14)|14]] || [[SmallGroup(64,14)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,15)|15]] || [[SmallGroup(64,15)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_8:C_8</math><br />
|-<br />
|64 || [[SmallGroup(64,16)|16]] || [[SmallGroup(64,16)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_8:C_8</math><br />
|-<br />
|64 || [[SmallGroup(64,17)|17]] || [[SmallGroup(64,17)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,18)|18]] || [[SmallGroup(64,18)]] || No || || || || || <math>(C_4 \times C_4):C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,19)|19]] || [[SmallGroup(64,19)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,20)|20]] || [[SmallGroup(64,20)]] || No || || || || || <math>(C_4 \times C_4):C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,21)|21]] || [[SmallGroup(64,21)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,22)|22]] || [[SmallGroup(64,22)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,23)|23]] || [[SmallGroup(64,23)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,24)|24]] || [[SmallGroup(64,24)]] || No || || || || || <math>M_{16}:C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,25)|25]] || [[SmallGroup(64,25)]] || No || || || || || <math>M_{16}:C_4</math><br />
|-<br />
|64 || [[C16xC4|26]] || [[C16xC4|<math>C_{16} \times C_4</math>]]|| <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|64 || [[SmallGroup(64,27)|27]] || [[SmallGroup(64,27)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_{16}:C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,28)|28]] || [[SmallGroup(64,28)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_{16}:C_4</math><br />
|-<br />
|64 || [[(C2xC2):C16|29]] || [[(C2xC2):C16|<math>(C_2)^2:C_{16}</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,30)|30]] || [[SmallGroup(64,30)]] || No || || || || || <math>M_{32}:C_2</math><br />
|-<br />
|64 || [[SmallGroup(64,31)|31]] || [[SmallGroup(64,31)]] || No || || || || || <math>M_{32}:C_2</math><br />
|-<br />
|64 || [[C2wrC4|32]] || [[(C2wrC4|<math>C_2 \wr C_4</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,33)|33]] || [[SmallGroup(64,33)]] || No || || || || || <br />
|-<br />
|64 || [[SmallGroup(64,34)|34]] || [[SmallGroup(64,31)]] || No || || || || || <math>(C_4 \times C_4):C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,35)|35]] || [[SmallGroup(64,35)]] || No || || || || || <math>(C_4 \times C_4):C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,36)|36]] || [[SmallGroup(64,36)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,37)|37]] || [[SmallGroup(64,37)]] || No || || || || || <math>C_4:Q_8</math>, fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,38)|38]] || [[SmallGroup(64,38)]] || No || || || || || <math>D_{16}:C_4</math>, fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,39)|39]] || [[SmallGroup(64,39)]] || No || || || || || <math>Q_{16}:C_2</math><br />
|-<br />
|64 || [[SmallGroup(64,40)|40]] || [[SmallGroup(64,40)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,41)|41]] || [[SmallGroup(64,41)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,42)|42]] || [[SmallGroup(64,42)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,43)|43]] || [[SmallGroup(64,43)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[C4:C16|44]] || [[C4:C16|<math>C_4:C_{16}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_4:C_{16}</math><br />
|-<br />
|64 || [[SmallGroup(64,45)|45]] || [[SmallGroup(64,45)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,46)|46]] || [[SmallGroup(64,46)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,47)|47]] || [[SmallGroup(64,47)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_{16}:C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,48)|48]] || [[SmallGroup(64,48)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_{16}:C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,48)|49]] || [[SmallGroup(64,49)]] || No || || || || ||<br />
|- <br />
|64 || [[C32xC2|50]] || [[C32xC2|<math>C_{32} \times C_2</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|64 || [[M6(2)|51]] || [[M6(2)|<math>M_6(2)</math>]] || No || || || || ||<br />
|-<br />
|64 || [[D64|52]] || [[D64|<math>D_{64}</math>]] || No || <math>k</math> || || || ||<br />
|-<br />
|64 || [[SD64|53]] || [[SD64|<math>SD_{64}</math>]] || No || <math>k</math> || || || ||<br />
|-<br />
|64 || [[Q64|54]] || [[Q64|<math>Q_{64}</math>]] || No || || || || ||<br />
|- <br />
|64 || [[C4xC4xC4|55]] || [[C4xC4xC4|<math>C_4 \times C_4 \times C_4</math>]] || <math>\mathcal{O}</math> ||4(4) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EL18a]]]||<br />
|-<br />
|64 || [[SmallGroup(64,56)|56]] || [[SmallGroup(64,56)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,57)|57]] || [[SmallGroup(64,57)]] || No || || || || ||<br />
|- <br />
|64 || [[C4x(C2xC2):C4|58]] || [[C4x(C2xC2):C4|<math>C_{4} \times (C_2 \times C_2):C_4</math>]] || || || || || || Fusion trivial?<br />
|- <br />
|64 || [[C4x(C4:C4)|59]] || [[C4x(C4:C4)|<math>C_{4} \times (C_4:C_4)</math>]] || || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,60)|60]] || [[SmallGroup(64,60)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,61)|61]] || [[SmallGroup(64,61)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,62)|62]] || [[SmallGroup(64,62)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,63)|63]] || [[SmallGroup(64,63)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|-<br />
|64 || [[SmallGroup(64,64)|64]] || [[SmallGroup(64,64)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,65)|65]] || [[SmallGroup(64,65)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,66)|66]] || [[SmallGroup(64,66)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,67)|67]] || [[SmallGroup(64,67)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,68)|68]] || [[SmallGroup(64,68)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|-<br />
|64 || [[SmallGroup(64,69)|69]] || [[SmallGroup(64,69)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,70)|70]] || [[SmallGroup(64,70)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|-<br />
|64 || [[SmallGroup(64,71)|71]] || [[SmallGroup(64,71)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,72)|72]] || [[SmallGroup(64,72)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|-<br />
|64 || [[SmallGroup(64,73)|73]] || [[SmallGroup(64,73)]] || No || || || || ||<br />
|-<br />
|64 || [[(C2)^3:Q8|74]] || [[(C2)^3:Q8|<math>(C_2)^3:Q_8</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,75)|75]] || [[SmallGroup(64,75)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,76)|76]] || [[SmallGroup(64,76)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|-<br />
|64 || [[SmallGroup(64,77)|77]] || [[SmallGroup(64,77)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,78)|78]] || [[SmallGroup(64,78)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,79)|79]] || [[SmallGroup(64,79)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|-<br />
|64 || [[SmallGroup(64,80)|80]] || [[SmallGroup(64,80)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,81)|81]] || [[SmallGroup(64,81)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|-<br />
|64 || [[SmallGroup(64,82)|82]] || [[SmallGroup(64,82)]] || <math>\mathcal{O}</math> || 6(6) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[Ea24]]] || Sylow 2-subgroup of <math>Sz(8)</math><br />
|-<br />
|64 || [[C8xC4xC2|83]] || [[C8xC4xC2|<math>C_{8} \times C_4 \times C_2</math>]]|| <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|64 || [[C2x(C8:C4)|84]] || [[C2x(C8:C4)|<math>C_{2} \times (C_8:C_4)</math>]]|| No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[M4(2)xC4|85]] || [[M4(2)xC4|<math>M_4(2) \times C_4</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,86)|86]] || [[SmallGroup(64,86)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[C2x(C2xC2):C8|87]] || [[C2x(C2xC2):C8|<math>C_{2} \times (C_2 \times C_2):C_8</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,88)|88]] || [[SmallGroup(64,88)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[(C8xC2xC2):C2|89]] || [[(C8xC2xC2):C2|<math>(C_8 \times C_2 \times C_2):C_2</math>]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[C2x(C2xC2xC2):C4|90]] || [[C2x(C2xC2xC2):C4|<math>C_2 \times (C_2 \times C_2 \times C_2):C_4</math>]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,91)|91]] || [[SmallGroup(64,91)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,92)|92]] || [[SmallGroup(64,92)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,93)|93]] || [[SmallGroup(64,93)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,94)|94]] || [[SmallGroup(64,94)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[C2x(D_8:C4)|95]] || [[C2x(D_8:C4)|<math>C_{2} \times (D_8:C_4)</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[C2x(Q_8:C4)|96]] || [[C2x(Q_8:C4)|<math>C_{2} \times (Q_8:C_4)</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,97)|97]] || [[SmallGroup(64,97)]] || No || || || || || <br />
|-<br />
|64 || [[SmallGroup(64,98)|98]] || [[SmallGroup(64,98)]] || No || || || || || <br />
|-<br />
|64 || [[SmallGroup(64,99)|99]] || [[SmallGroup(64,99)]] || No || || || || || <br />
|-<br />
|64 || [[SmallGroup(64,100)|100]] || [[SmallGroup(64,100)]] || No || || || || || <br />
|-<br />
|64 || [[C2x(C4wrC2)|101]] || [[C2x(C4wrC2)|<math>C_{2} \times (C_4 \wr C_2)</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[(C4xC4)(C2:C2)|102]] || [[(C4xC4)(C2:C2)|<math>(C_4 \times C_4):(C_2 \times C_2)</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[C2x(C4:C8)|103]] || [[C2x(C4:C8)|<math>C_{2} \times (C_4:C_8)</math>]]|| No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[C4:M4(2)|104]] || [[C4:M4(2)|<math>C_{4}:M_4(2)</math>]]|| No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,105)|105]] || [[SmallGroup(64,105)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,106)|106]] || [[SmallGroup(64,106)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,245)|245]] || [[SmallGroup(64,245)]] || <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[Ea24]]] || Sylow 2-subgroup of <math>PSU_3(4)</math><br />
|}--><br />
<br />
==Blocks for <math>p=3</math>==<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>1 \leq |D| \leq 27</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|- <br />
| 3 || [[C3|1]] || [[C3|<math>C_3</math>]] || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
|9 || [[C9|1]] ||[[C9|<math>C_9</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|-<br />
|9 || [[C3xC3|2]] || [[C3xC3|<math>C_3 \times C_3</math>]] || || || || ||<br />
|-<br />
|27 || [[C27|1]] || [[C27|<math>C_{27}</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|27 || [[C9xC3|2]] || [[C9xC3|<math>C_9 \times C_3</math>]] || || || || ||<br />
|-<br />
|27 || [[3_+^3|3]] || [[3_+^3|<math>3_+^{1+2}</math>]] || || || || ||<br />
|-<br />
|27 || [[3_-^3|4]] || [[3_-^3|<math>3_-^{1+2}</math>]] || || || || ||<br />
|-<br />
|27 || [[C3xC3xC3|5]] || [[C3xC3xC3|<math>C_3 \times C_3 \times C_3</math>]] || || || || ||<br />
|}<br />
<br />
==Blocks for <math>p=5</math>==<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>5 \leq |D| \leq 125</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|- <br />
|5 || [[C5|1]] || [[C5|<math>C_5</math>]] ||6(6) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
|25 || [[C25|1]] ||[[C25|<math>C_{25}</math>]] || 6(6) || No || <math>\mathcal{O}</math> || || Max 12 classes <br />
|-<br />
|25 || [[C5xC5|2]] || [[C5xC5|<math>C_5 \times C_5</math>]] || || || || ||<br />
|- <br />
|125 || [[C125|1]] ||[[C125|<math>C_{125}</math>]] || || || || || <br />
|-<br />
|125 || [[C25xC5|2]] || [[C25xC5|<math>C_{25} \times C_5</math>]] || || || || || <br />
|-<br />
|125 || [[5_+^3|3]] || [[5_+^3|<math>5_+^{1+2}</math>]] || 62(62) || <math>\mathcal{O}</math> || || [[References#A|[AE23]]] || Inertial quotients are consistent within classes<br />
|-<br />
|125 || [[5_-^3|4]] || [[5_-^3|<math>5_-^{1+2}</math>]] || || || || ||<br />
|-<br />
|125 || [[C5xC5xC5|5]] || [[C5xC5xC5|<math>C_5 \times C_5 \times C_5</math>]] || || || || ||<br />
|}<br />
<br />
==Blocks for <math>p\geq 7</math>==<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>|D|</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|- <br />
|7 || [[C7|1]] || [[C7|<math>C_7</math>]] ||14(14) ||No || <math>\mathcal{O}</math> || ||Max 21 classes <br />
|- <br />
|11|| [[C11|1]] || [[C11|<math>C_{11}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|- <br />
|13 || [[C13|1]] || [[C13|<math>C_{13}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|-<br />
|17|| [[C17|1]] || [[C17|<math>C_{17}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|- <br />
|19 || [[C19|1]] || [[C19|<math>C_{19}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|-<br />
|23 || [[C23|1]] || [[C23|<math>C_{23}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|}</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Classification_by_p-group&diff=1237
Classification by p-group
2024-01-09T17:36:39Z
<p>Charles Eaton: /* Blocks for p=2 */</p>
<hr />
<div>'''Classification of Morita equivalences for blocks with a given defect group'''<br />
<br />
On this page we list classifications of Morita equivalence classes for each isomorphism class of p-groups in turn. [[Generic classifications by p-group class|Generic classifications for classes of p-groups can be found here]].<br />
<br />
See [[Labelling for Morita equivalence classes|this page]] for a description of the labelling conventions.<br />
<br />
== Blocks for <math> p=2 </math> ==<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>1 \leq |D| \leq 8</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|- <br />
| 2 || [[C2|1]] || [[C2|<math>C_2</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
| 4 || [[C4|1]] || [[C4|<math>C_4</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
| 4 || [[C2xC2|2]] || [[C2xC2|<math>C_2 \times C_2</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Er82], [Li94] ]] ||<br />
|- <br />
|8 || [[C8|1]] || [[C8|<math>C_8</math>]] ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|8 || [[C4xC2|2]] || [[C4xC2|<math>C_4 \times C_2</math>]] ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|8 || [[D8|3]] || [[D8|<math>D_8</math>]] ||6(?) || <math>k</math> || <math>k</math> || [[References|[Er87] ]] || Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] <br />
|-<br />
|8 || [[Q8|4]] || [[Q8|<math>Q_8</math>]] ||3(3) || <math>\mathcal{O}</math> || <math>k</math> || [[References|[Er88a], [Er88b], [HKL07], [Ei16]]] || <br />
|-<br />
|8 || [[C2xC2xC2|5]] || [[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]] || 8(8) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References| [Ea16]]] || Uses CFSG<br />
|}<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>|D|=16</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Donovan (w.r.t)?<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|16 || [[C16|1]] || [[C16|<math>C_{16}</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|16 || [[C4xC4|2]] || [[C4xC4|<math>C_4 \times C_4</math>]] || <math>\mathcal{O}</math> || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[EKKS14] ]] ||<br />
|-<br />
|16 || [[MNA(2,1)|3]] || [[MNA(2,1)]] || No || 3(?) || No || || [[References|[Sa11] ]] || Block invariants known<br />
|-<br />
|16 || [[C4:C4|4]] || [[C4:C4|<math>C_4:C_4</math>]] || <math>\mathcal{O}</math>|| 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[CG12], [Sa12b]]] ||<br />
|-<br />
|16 || [[C8xC2|5]] || [[C8xC2|<math>C_8 \times C_2</math>]]|| <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|16 || [[M16|6]] || [[M16|<math>M_{16}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[CG12], [Sa12b] ]] || <br />
|-<br />
|16 || [[D16|7]] || [[D16|<math>D_{16}</math>]] || <math>k</math>|| 5(?) || <math>k</math> || <math>k</math> || [[References|[Er87]]] || Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] <br />
|-<br />
|16 || [[SD16|8]] || [[SD16|<math>SD_{16}</math>]] || <math>k</math> || 7(?) || || || [[References|[Er88c], [Er90b]]] || Two other possible classes<br />
|-<br />
|16 || [[Q16|9]] || [[Q16|<math>Q_{16}</math>]] || No || 6(?) || || <math>k</math> || [[References|[Er88a], [Er88b], [Ho97]]] || Two possibly infinite families when <math>l(B)=2</math>. Classified over <math>\mathcal{O}</math> when <math>l(B)=3</math> in [[References#E|[Ei16]]]. Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20b]]]<br />
|-<br />
|16 || [[C4xC2xC2|10]] || [[C4xC2xC2|<math>C_4 \times C_2 \times C_2</math>]]|| <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[EL18a]]] ||<br />
|-<br />
|16 || [[D8xC2|11]] || [[D8xC2|<math>D_8 \times C_2</math>]] || No || || || || [[References|[Sa12] ]] || Block invariants known<br />
|-<br />
|16 || [[Q8xC2|12]] || [[Q8xC2|<math>Q_8 \times C_2</math>]] || <math>\mathcal{O}</math> || 3(3) || No || || [[References#E|[EL20]]] || Block invariants known by [[References#S|[Sa13]]]<br />
|-<br />
|16 || [[D8*C4|13]] || [[D8*C4|<math>D_8*C_4</math>]] || No || 3(?) || No || || [[References|[Sa13b] ]] || Block invariants known<br />
|-<br />
|16 || [[(C2)^4|14]] || [[(C2)^4|<math>(C_2)^4</math>]] || <math>\mathcal{O}</math> || 16(16) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Ea18] ]] ||<br />
|}<br />
<br />
The table for defect groups of order 32 takes as its starting point Table 13.1 of Sambale's book [[References|[Sa14]]].<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>|D|=32</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Donovan (w.r.t)?<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|32 || [[C32|1]] || [[C32|<math>C_{32}</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|32 || [[MNA(2,2)|2]] || [[MNA(2,2)|<math>MNA(2,2)</math>]] || <math>\mathcal{O}</math> || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EKS12]]] ||<br />
|-<br />
|32 || [[C8xC4|3]] || [[C8xC4|<math>C_8 \times C_4</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|-<br />
|32 || [[C8:C4|4]] || [[C8:C4|<math>C_8:C_4</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[MNA(3,1)|5]] || [[MNA(3,1)|<math>MNA(3,1)</math>]] || No || || || || [[References#S|[Sa11] ]] || Invariants known<br />
|-<br />
|32 || [[MNA(2,1):C2|6]] || [[MNA(3,1):C2|<math>MNA(2,1):C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,7)|7]] || [[SmallGroup(32,7)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] || <math>M_{16}:C_2</math><br />
|-<br />
|32 || [[2.MNA(2,1)|8]] || [[2.MNA(2,1)|<math>2.MNA(2,1)</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[D8:C4|9]] || [[D8:C4|<math>D_8:C_4</math>]] || No || || || || [[References#S|[Sa14,10.23]]] || Invariants known<br />
|-<br />
|32 || [[Q8:C4|10]] || [[Q8:C4|<math>Q_8:C_4</math>]] || No || || || || [[References#S|[Sa14,10.25]]] || Invariants known<br />
|-<br />
|32 || [[C4wrC2|11]] || [[C4wrC2|<math>C_4 \wr C_2</math>]] || No || 6(6) || No || || [[References#K|[Ku80]]], [[References#K|[KoLaSa23]]] || Invariants known. Principal blocks classified up to source algebra equivalence in [[References#K|[KoLaSa23]]]<br />
|-<br />
|32 || [[C4:C8|12]] || [[C4:C8|<math>C_4:C_8</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[C8:C4a|13]] || [[C8:C4a|<math>C_8:C_4=\langle a,b|a^8=b^4=1, ba=a^3b \rangle</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[C8:C4b|14]] || [[C8:C4b|<math>C_8:C_4=\langle a,b|a^8=b^4=1, ba=a^7b \rangle</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,15)|15]] || [[SmallGroup(32,15)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[C16xC2|16]] || [[C16xC2|<math>C_{16} \times C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|32 || [[M32|17]] || [[M32|<math>M_{32}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[D32|18]] || [[D32|<math>D_{32}</math>]] || <math>k</math> || 5(?) || <math>k</math> || <math>k</math> || [[References#E|[Er87]]] || Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] <br />
|-<br />
|32 || [[SD32|19]] || [[SD32|<math>SD_{32}</math>]] || <math>k</math> || || || || ||<br />
|-<br />
|32 || [[Q32|20]] || [[Q32|<math>Q_{32}</math>]] || No || || || || [[References#E|[Er88a], [Er88b], [Ho97]]] || Two possibly infinite families when <math>l(B)=2</math>. Classified over <math>\mathcal{O}</math> when <math>l(B)=3</math> in [[References#E|[Ei16]]]. Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20b]]]<br />
|-<br />
|32 || [[C4xC4xC2|21]] || [[C4xC4xC2|<math>C_4 \times C_4 \times C_2</math>]] || <math>\mathcal{O}</math> || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EKKS14]]] <br />
|-<br />
|32 || [[MNA(2,1)xC2|22]] || [[MNA(2,1)xC2|<math>MNA(2,1) \times C_2</math>]] || No || || || || [[References#S|[Sa14,10.25]]] || Invariants known<br />
|-<br />
|32 || [[(C4:C4)xC2|23]] || [[(C4:C4)xC2|<math>(C_4:C_4) \times C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,24)|24]] || [[SmallGroup(32,24)]]<!--<math>(C_4 \times C_4):C_2=\langle a,b,c \mid a^4 = b^4 = c^2 = e, ab = ba, ac = ca, cb = a^2bc \rangle</math>]]--> || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[D8xC4|25]] || [[D8xC4|<math>D_8 \times C_4</math>]] || No || || || || [[References#S|[Sa14,9.7]]] ||<br />
Invariants known<br />
|-<br />
|32 || [[Q8xC4|26]] || [[Q8xC4|<math>Q_8 \times C_4</math>]] || <math>\mathcal{O}</math> || 3(3) || No || || [[References#E|[EL20]]] || Invariants known by [[References#S|[Sa14,9.28]]]<br />
|-<br />
|32 || [[SmallGroup(32,27)|27]] || [[SmallGroup(32,27)]]<!--|<math>(C_4 \times C_4):C_2=\langle x,y,z,a,b \mid x^2 = y^2 = z^2 = a^2 = b^2 = e, xy = yx, xz, = zx, yz = zy, aza^{-1} = xz, bzb^{-1} = yz, ax = xa, ay = ya, bx = xb, by = yb \rangle</math>]]--> || No || || || || ||<br />
|-<br />
|32 || [[SmallGroup(32,28)|28]] || [[SmallGroup(32,28)]] || No || || || || [[References#S|[Sa14,13.11]]] || Invariants known<br />
|-<br />
|32 || [[SmallGroup(32,29)|29]] || [[SmallGroup(32,29)]] || No || || || || [[References#S|[Sa14,13.11]]] || Invariants known<br />
|-<br />
|32 || [[SmallGroup(32,30)|30]] || [[SmallGroup(32,30)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,31)|31]] || [[SmallGroup(32,31)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,32)|32]] || [[SmallGroup(32,32)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,33)|33]] || [[SmallGroup(32,33)]] || No || || || || [[References|[Sa14,13.12]]] || Invariants partly known<br />
|-<br />
|32 || [[SmallGroup(32,34)|34]] || [[SmallGroup(32,34)]] || No || || || || [[References|[Sa14,13.12]]] || Invariants partly known<br />
|-<br />
|32 || [[C4:Q8|35]] || [[C4:Q8|<math>C_4:Q_8</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[C8xC2xC2|36]] || [[C8xC2xC2|<math>C_8 \times C_2 \times C_2</math>]] || <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EL18a]]] ||<br />
|-<br />
|32 || [[M16xC2|37]] || [[M16xC2|<math>M_{16} \times C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[D8*C8|38]] || [[D8*C8|<math>D_8*C_8</math>]] || No || || || || [[References#S|[Sa14,9.18]]] || Invariants known<br />
|-<br />
|32 || [[D16xC2|39]] || [[D16xC2|<math>D_{16} \times C_2</math>]] || No || || || || [[References#S|[Sa14,9.7]]] || Invariants known<br />
|-<br />
|32 || [[SD16xC2|40]] || [[SD16xC2|<math>SD_{16} \times C_2</math>]] || No || || || || [[References#S|[Sa14,9.37]]] || Invariants known<br />
|-<br />
|32 || [[Q16xC2|41]] || [[Q16xC2|<math>Q_{16} \times C_2</math>]] || No || || || || [[References#S|[Sa14,9.28]]] || Invariants known<br />
|-<br />
|32 || [[D16*C4|42]] || [[D16*C4|<math>D_{16}*C_4</math>]] || No || || || || [[References|[Sa14,9.18]]] || Invariants known<br />
|-<br />
|32 || [[SmallGroup(32,43)|43]] || [[SmallGroup(32,43)]] || No || || || || ||<br />
|-<br />
|32 || [[SmallGroup(32,44)|44]] || [[SmallGroup(32,44)]] || No || || || || ||<br />
|-<br />
|32 || [[C4xC2xC2xC2|45]] || [[C4xC2xC2xC2|<math>C_4 \times C_2 \times C_2 \times C_2</math>]] || <math>\mathcal{O}</math> || 8(8) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EL23]]] ||<br />
|-<br />
|32 || [[D8xC2xC2|46]] || [[D8xC2xC2|<math>D_8 \times C_2 \times C_2</math>]] || No || || || || ||<br />
|-<br />
|32 || [[Q8xC2xC2|47]] || [[Q8xC2xC2|<math>Q_8 \times C_2 \times C_2</math>]] || No || || || || ||<br />
|-<br />
|32 || [[D8*C4xC2|48]] || [[D8*C4xC2|<math>(D_8*C_4) \times C_2</math>]] || No || || || || ||<br />
|-<br />
|32 || [[D8*D8|49]] || [[D8*D8|<math>D_8*D_8</math>]] || No || || || || [[References#S|[Sa13c]]] || Invariants partly known<br />
|-<br />
|32 || [[D8*Q8|50]] || [[D8*Q8|<math>D_8*Q_8</math>]] || No || || || || [[References#S|[Sa13c]]] || Invariants partly known<br />
|-<br />
|32 || [[(C2)^5|51]] || [[(C2)^5|<math>(C_2)^5</math>]] || <math>\mathcal{O}</math> || 34 (34) || <math>\mathcal{O}</math> || || [[References#A|[Ar19]]] || Derived eq. classes determined for 30 of the 34 Morita eq. classes. <br />
|}<br />
<br />
<!--<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>|D|=64</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Donovan (w.r.t)?<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|64 || [[C64|1]] || [[C64|<math>C_{64}</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
|64 || [[C8xC8|2]] || [[C8xC8|<math>C_8 \times C_8</math>]] || <math>\mathcal{O}</math> ||2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EKKS14]]]||<br />
|-<br />
|64 || [[SmallGroup(64,3)|3]] || [[SmallGroup(64,3)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_8:C_8</math><br />
|-<br />
|64 || [[(C2xC2xC2):C8|4]] || [[(C2xC2xC2):C8|<math>(C_2)^3:C_8</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,5)|5]] || [[SmallGroup(64,5)]] || No || || || || ||<br />
|-<br />
|64 || [[(D8:C8|6]] || [[D8:C8|<math>D_8:C_8</math>]] || No || || || || ||<br />
|-<br />
|64 || [[(Q8:C8|7]] || [[Q8:C8|<math>Q_8:C_8</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,8)|8]] || [[SmallGroup(64,8)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,9)|9]] || [[SmallGroup(64,9)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,10)|10]] || [[SmallGroup(64,10)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,11)|11]] || [[SmallGroup(64,11)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,12)|12]] || [[SmallGroup(64,12)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,13)|13]] || [[SmallGroup(64,13)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,14)|14]] || [[SmallGroup(64,14)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,15)|15]] || [[SmallGroup(64,15)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_8:C_8</math><br />
|-<br />
|64 || [[SmallGroup(64,16)|16]] || [[SmallGroup(64,16)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_8:C_8</math><br />
|-<br />
|64 || [[SmallGroup(64,17)|17]] || [[SmallGroup(64,17)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,18)|18]] || [[SmallGroup(64,18)]] || No || || || || || <math>(C_4 \times C_4):C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,19)|19]] || [[SmallGroup(64,19)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,20)|20]] || [[SmallGroup(64,20)]] || No || || || || || <math>(C_4 \times C_4):C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,21)|21]] || [[SmallGroup(64,21)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,22)|22]] || [[SmallGroup(64,22)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,23)|23]] || [[SmallGroup(64,23)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,24)|24]] || [[SmallGroup(64,24)]] || No || || || || || <math>M_{16}:C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,25)|25]] || [[SmallGroup(64,25)]] || No || || || || || <math>M_{16}:C_4</math><br />
|-<br />
|64 || [[C16xC4|26]] || [[C16xC4|<math>C_{16} \times C_4</math>]]|| <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|64 || [[SmallGroup(64,27)|27]] || [[SmallGroup(64,27)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_{16}:C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,28)|28]] || [[SmallGroup(64,28)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_{16}:C_4</math><br />
|-<br />
|64 || [[(C2xC2):C16|29]] || [[(C2xC2):C16|<math>(C_2)^2:C_{16}</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,30)|30]] || [[SmallGroup(64,30)]] || No || || || || || <math>M_{32}:C_2</math><br />
|-<br />
|64 || [[SmallGroup(64,31)|31]] || [[SmallGroup(64,31)]] || No || || || || || <math>M_{32}:C_2</math><br />
|-<br />
|64 || [[C2wrC4|32]] || [[(C2wrC4|<math>C_2 \wr C_4</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,33)|33]] || [[SmallGroup(64,33)]] || No || || || || || <br />
|-<br />
|64 || [[SmallGroup(64,34)|34]] || [[SmallGroup(64,31)]] || No || || || || || <math>(C_4 \times C_4):C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,35)|35]] || [[SmallGroup(64,35)]] || No || || || || || <math>(C_4 \times C_4):C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,36)|36]] || [[SmallGroup(64,36)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,37)|37]] || [[SmallGroup(64,37)]] || No || || || || || <math>C_4:Q_8</math>, fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,38)|38]] || [[SmallGroup(64,38)]] || No || || || || || <math>D_{16}:C_4</math>, fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,39)|39]] || [[SmallGroup(64,39)]] || No || || || || || <math>Q_{16}:C_2</math><br />
|-<br />
|64 || [[SmallGroup(64,40)|40]] || [[SmallGroup(64,40)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,41)|41]] || [[SmallGroup(64,41)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,42)|42]] || [[SmallGroup(64,42)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,43)|43]] || [[SmallGroup(64,43)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[C4:C16|44]] || [[C4:C16|<math>C_4:C_{16}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_4:C_{16}</math><br />
|-<br />
|64 || [[SmallGroup(64,45)|45]] || [[SmallGroup(64,45)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,46)|46]] || [[SmallGroup(64,46)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,47)|47]] || [[SmallGroup(64,47)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_{16}:C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,48)|48]] || [[SmallGroup(64,48)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_{16}:C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,48)|49]] || [[SmallGroup(64,49)]] || No || || || || ||<br />
|- <br />
|64 || [[C32xC2|50]] || [[C32xC2|<math>C_{32} \times C_2</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|64 || [[M6(2)|51]] || [[M6(2)|<math>M_6(2)</math>]] || No || || || || ||<br />
|-<br />
|64 || [[D64|52]] || [[D64|<math>D_{64}</math>]] || No || <math>k</math> || || || ||<br />
|-<br />
|64 || [[SD64|53]] || [[SD64|<math>SD_{64}</math>]] || No || <math>k</math> || || || ||<br />
|-<br />
|64 || [[Q64|54]] || [[Q64|<math>Q_{64}</math>]] || No || || || || ||<br />
|- <br />
|64 || [[C4xC4xC4|55]] || [[C4xC4xC4|<math>C_4 \times C_4 \times C_4</math>]] || <math>\mathcal{O}</math> ||4(4) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EL18a]]]||<br />
|-<br />
|64 || [[SmallGroup(64,56)|56]] || [[SmallGroup(64,56)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,57)|57]] || [[SmallGroup(64,57)]] || No || || || || ||<br />
|- <br />
|64 || [[C4x(C2xC2):C4|58]] || [[C4x(C2xC2):C4|<math>C_{4} \times (C_2 \times C_2):C_4</math>]] || || || || || || Fusion trivial?<br />
|- <br />
|64 || [[C4x(C4:C4)|59]] || [[C4x(C4:C4)|<math>C_{4} \times (C_4:C_4)</math>]] || || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,60)|60]] || [[SmallGroup(64,60)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,61)|61]] || [[SmallGroup(64,61)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,62)|62]] || [[SmallGroup(64,62)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,63)|63]] || [[SmallGroup(64,63)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,64)|64]] || [[SmallGroup(64,64)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,65)|65]] || [[SmallGroup(64,65)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,66)|66]] || [[SmallGroup(64,66)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,67)|67]] || [[SmallGroup(64,67)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,68)|68]] || [[SmallGroup(64,68)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|-<br />
|64 || [[SmallGroup(64,69)|69]] || [[SmallGroup(64,69)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,70)|70]] || [[SmallGroup(64,70)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|-<br />
|64 || [[SmallGroup(64,71)|71]] || [[SmallGroup(64,71)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,72)|72]] || [[SmallGroup(64,72)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|-<br />
|64 || [[SmallGroup(64,73)|73]] || [[SmallGroup(64,73)]] || No || || || || ||<br />
|-<br />
|64 || [[(C2)^3:Q8|74]] || [[(C2)^3:Q8|<math>(C_2)^3:Q_8</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,75)|75]] || [[SmallGroup(64,75)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,76)|76]] || [[SmallGroup(64,76)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|-<br />
|64 || [[SmallGroup(64,77)|77]] || [[SmallGroup(64,77)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,78)|78]] || [[SmallGroup(64,78)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,79)|79]] || [[SmallGroup(64,79)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|-<br />
|64 || [[SmallGroup(64,80)|80]] || [[SmallGroup(64,80)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,81)|81]] || [[SmallGroup(64,81)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|-<br />
|64 || [[SmallGroup(64,82)|82]] || [[SmallGroup(64,82)]] || <math>\mathcal{O}</math> || 6(6) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[Ea24]]] || Sylow 2-subgroup of <math>Sz(8)</math><br />
|-<br />
|64 || [[C8xC4xC2|83]] || [[C8xC4xC2|<math>C_{8} \times C_4 \times C_2</math>]]|| <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|64 || [[C2x(C8:C4)|84]] || [[C2x(C8:C4)|<math>C_{2} \times (C_8:C_4)</math>]]|| No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[M4(2)xC4|85]] || [[M4(2)xC4|<math>M_4(2) \times C_4</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,86)|86]] || [[SmallGroup(64,86)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[C2x(C2xC2):C8|87]] || [[C2x(C2xC2):C8|<math>C_{2} \times (C_2 \times C_2):C_8</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,88)|88]] || [[SmallGroup(64,88)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[(C8xC2xC2):C2|89]] || [[(C8xC2xC2):C2|<math>(C_8 \times C_2 \times C_2):C_2</math>]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[C2x(C2xC2xC2):C4|90]] || [[C2x(C2xC2xC2):C4|<math>C_2 \times (C_2 \times C_2 \times C_2):C_4</math>]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,91)|91]] || [[SmallGroup(64,91)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,92)|92]] || [[SmallGroup(64,92)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,93)|93]] || [[SmallGroup(64,93)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,94)|94]] || [[SmallGroup(64,94)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[C2x(D_8:C4)|95]] || [[C2x(D_8:C4)|<math>C_{2} \times (D_8:C_4)</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[C2x(Q_8:C4)|96]] || [[C2x(Q_8:C4)|<math>C_{2} \times (Q_8:C_4)</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,97)|97]] || [[SmallGroup(64,97)]] || No || || || || || <br />
|-<br />
|64 || [[SmallGroup(64,98)|98]] || [[SmallGroup(64,98)]] || No || || || || || <br />
|-<br />
|64 || [[SmallGroup(64,99)|99]] || [[SmallGroup(64,99)]] || No || || || || || <br />
|-<br />
|64 || [[SmallGroup(64,100)|100]] || [[SmallGroup(64,100)]] || No || || || || || <br />
|-<br />
|64 || [[C2x(C4wrC2)|101]] || [[C2x(C4wrC2)|<math>C_{2} \times (C_4 \wr C_2)</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[(C4xC4)(C2:C2)|102]] || [[(C4xC4)(C2:C2)|<math>(C_4 \times C_4):(C_2 \times C_2)</math>]]|| No || || || || ||<br />
|-<br />
|64 || [[C2x(C4:C8)|103]] || [[C2x(C4:C8)|<math>C_{2} \times (C_4:C_8)</math>]]|| No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[C4:M4(2)|104]] || [[C4:M4(2)|<math>C_{4}:M_4(2)</math>]]|| No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,105)|105]] || [[SmallGroup(64,105)]] || No || || || || || Fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,106)|106]] || [[SmallGroup(64,106)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,245)|245]] || [[SmallGroup(64,245)]] || <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[Ea24]]] || Sylow 2-subgroup of <math>PSU_3(4)</math><br />
|}--><br />
<br />
==Blocks for <math>p=3</math>==<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>1 \leq |D| \leq 27</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|- <br />
| 3 || [[C3|1]] || [[C3|<math>C_3</math>]] || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
|9 || [[C9|1]] ||[[C9|<math>C_9</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|-<br />
|9 || [[C3xC3|2]] || [[C3xC3|<math>C_3 \times C_3</math>]] || || || || ||<br />
|-<br />
|27 || [[C27|1]] || [[C27|<math>C_{27}</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|27 || [[C9xC3|2]] || [[C9xC3|<math>C_9 \times C_3</math>]] || || || || ||<br />
|-<br />
|27 || [[3_+^3|3]] || [[3_+^3|<math>3_+^{1+2}</math>]] || || || || ||<br />
|-<br />
|27 || [[3_-^3|4]] || [[3_-^3|<math>3_-^{1+2}</math>]] || || || || ||<br />
|-<br />
|27 || [[C3xC3xC3|5]] || [[C3xC3xC3|<math>C_3 \times C_3 \times C_3</math>]] || || || || ||<br />
|}<br />
<br />
==Blocks for <math>p=5</math>==<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>5 \leq |D| \leq 125</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|- <br />
|5 || [[C5|1]] || [[C5|<math>C_5</math>]] ||6(6) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
|25 || [[C25|1]] ||[[C25|<math>C_{25}</math>]] || 6(6) || No || <math>\mathcal{O}</math> || || Max 12 classes <br />
|-<br />
|25 || [[C5xC5|2]] || [[C5xC5|<math>C_5 \times C_5</math>]] || || || || ||<br />
|- <br />
|125 || [[C125|1]] ||[[C125|<math>C_{125}</math>]] || || || || || <br />
|-<br />
|125 || [[C25xC5|2]] || [[C25xC5|<math>C_{25} \times C_5</math>]] || || || || || <br />
|-<br />
|125 || [[5_+^3|3]] || [[5_+^3|<math>5_+^{1+2}</math>]] || 62(62) || <math>\mathcal{O}</math> || || [[References#A|[AE23]]] || Inertial quotients are consistent within classes<br />
|-<br />
|125 || [[5_-^3|4]] || [[5_-^3|<math>5_-^{1+2}</math>]] || || || || ||<br />
|-<br />
|125 || [[C5xC5xC5|5]] || [[C5xC5xC5|<math>C_5 \times C_5 \times C_5</math>]] || || || || ||<br />
|}<br />
<br />
==Blocks for <math>p\geq 7</math>==<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>|D|</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|- <br />
|7 || [[C7|1]] || [[C7|<math>C_7</math>]] ||14(14) ||No || <math>\mathcal{O}</math> || ||Max 21 classes <br />
|- <br />
|11|| [[C11|1]] || [[C11|<math>C_{11}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|- <br />
|13 || [[C13|1]] || [[C13|<math>C_{13}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|-<br />
|17|| [[C17|1]] || [[C17|<math>C_{17}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|- <br />
|19 || [[C19|1]] || [[C19|<math>C_{19}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|-<br />
|23 || [[C23|1]] || [[C23|<math>C_{23}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|}</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=References&diff=1236
References
2024-01-09T11:20:24Z
<p>Charles Eaton: [Ea24] and [Hi63]</p>
<hr />
<div>[[#A|A,]] [[#B|B,]] [[#C|C,]] [[#D|D,]] [[#E|E,]] [[#F|F,]] [[#G|G,]] [[#H|H,]] [[#I|I,]] [[#J|J,]] [[#K|K,]] [[#L|L,]] [[#M|M,]] [[#N|N,]] [[#O|O,]] [[#P|P,]] [[#Q|Q,]] [[#R|R,]] [[#S|S,]] [[#T|T]] [[#U|U,]] [[#V|V,]] [[#W|W,]] [[#X|X,]] [[#Y|Y,]] [[#Z|Z,]] <br />
<br />
{| <br />
|- id="A"<br />
|[Al79] || '''J. L. Alperin''', ''Projective modules for <math>SL(2,2^n)</math>'', J. Pure and Applied Algebra '''15''' (1979), 219-234.<br />
|-<br />
|[Al80] || '''J. L. Alperin''', ''Local representation theory'', The Santa Cruz Conference on Finite Groups., Proc. Sympos. Pure Math. '''37''' (1980), 369-375.<br />
|-<br />
|[AE81] || '''J. L. Alperin and L. Evens''', ''Representations, resoluutions and Quillen's dimension theorem'', J. Pure Appl. Algebra '''22''' (1981), 1-9.<br />
|-<br />
|[AE04] || '''Jianbei An and C. W. Eaton''', ''Blocks with trivial intersection defect groups'', Math. Z. '''247''' (2004), 461-486.<br />
|-<br />
|[AE04] || '''Jianbei An and C. W. Eaton''', [https://arxiv.org/abs/2310.02150 ''Morita equivalence classes of blocks with extraspecial defect groups <math>p_+^{1+2}</math>''], [https://arxiv.org/abs/2310.02150 arxiv:2310.02150]<br />
|-<br />
|[Ar19] || '''C. G. Ardito''', [https://arxiv.org/abs/1908.02652 ''Morita equivalence classes of blocks with elementary abelian defect groups of order 32''], J. Algebra '''573''' (2021), 297-335.<br />
|-<br />
|[ArMcK20] || '''C. G. Ardito and E. McKernon''', ''[https://arxiv.org/abs/2010.08329 ''2-blocks with an abelian defect group and a freely acting cyclic inertial quotient''], [https://arxiv.org/abs/2010.08329 arxiv.org/abs/2010.08329]<br />
|-<br />
|[AS20] || '''C. G. Ardito and B. Sambale''', [http://www.advgrouptheory.com/journal/Volumes/12/ArditoSambale.pdf ''Broué's Conjecture for 2-blocks with elementary abelian defect groups of order 32''], Advances in Group Theory and Applications 12 (2021), 71–78. <br />
|-<br />
|[AKO11] || '''M. Aschbacher, R. Kessar and B. Oliver''', ''Fusion systems in algebra and topology'', London Mathematical Society Lecture Notes '''391''', Cambridge University Press (2011).<br />
|- id="B"<br />
|[BK07] || '''D. Benson and R. Kessar''', ''Blocks inequivalent to their Frobenius twists'', J. Algebra '''315''' (2007), 588-599.<br />
|-<br />
|[BS23] || '''D. Benson and B. Sambale''', [https://arxiv.org/abs/2301.10537 ''Finite dimensional algebras not arising as blocks in group algebras''], [https://arxiv.org/pdf/2301.10537 arxiv:2301.10537]<br />
|-<br />
|[BKL18] || '''R. Boltje, R. Kessar, and M. Linckelmann''', [https://doi.org/10.1016/j.jalgebra.2019.02.045 ''On Picard groups of blocks of finite groups''], J. Algebra '''558''' (2020), 70-101.<br />
|-<br />
|[Bra41] || '''R. Brauer''', ''Investigations on group characters'', Ann. Math. '''42''' (1941), 936-958.<br />
|-<br />
|[BP80] || '''M. Broué and L. Puig''', ''A Frobenius theorem for blocks'', Invent. Math. '''56''' (1980), 117-128.<br />
|-<br />
|[BP80b] || '''M. Broué and L. Puig''', ''Characters and local structure in G-algebras'', J. Algebra '''63''' (1980), 306-317.<br />
|- id="C"<br />
|[Cr11] || '''D. A. Craven''', ''The Theory of Fusion Systems: An Algebraic Approach'', Cambridge University Press (2011).<br />
|-<br />
|[Cr12] || '''D. A. Craven''', [https://arxiv.org/abs/1207.0116 ''Perverse Equivalences and Broué's Conjecture II: The Cyclic Case''], [https://arxiv.org/abs/1207.0116 arXiv:1207.0116]<br />
|-<br />
|[CDR18] || '''D. A. Craven, O. Dudas and R. Rouquier''', [https://arxiv.org/abs/1701.07097 ''The Brauer trees of unipotent blocks''], to appear, J. EMS, [https://arxiv.org/abs/1701.07097 arXiv:1701.07097] <br />
|-<br />
|[CEKL11] || '''D. A. Craven, C. W. Eaton, R. Kessar and M. Linckelmann''', ''The structure of blocks with a Klein four defect group'', Math. Z. '''268''' (2011), 441-476.<br />
|-<br />
|[CG12] || '''D. A. Craven and A. Glesser''', ''Fusion systems on small p-groups'', Trans. AMS '''364''' (2012) 5945-5967.<br />
|-<br />
|[CR13] || '''D. A. Craven and R. Rouquier''', ''Perverse equivalences and Broué's conjecture'', Adv. Math. '''248''' (2013), 1-58.<br />
|-<br />
|[CuRe81a] || '''C. W. Curtis and I. Reiner''', ''Methods of representation theory, with applications to finite groups and orders, Volume I'', Wiley-Interscience (1981).<br />
|-<br />
|[CuRe81b] || '''C. W. Curtis and I. Reiner''', ''Methods of representation theory, with applications to finite groups and orders, Volume II'', Wiley-Interscience (1981).<br />
|- id="D"<br />
|[Da66] || '''E. C. Dade''', ''Blocks with cyclic defect groups'', Ann. Math. '''84''' (1966), 20-48. <br />
|-<br />
|[DE20] || '''S. Danz and K. Erdmann''', [https://arxiv.org/abs/2008.10999 ''On Ext-Quivers of Blocks of weight two for symmetric groups''], [https://arxiv.org/abs/2008.10999 arXiv:2008.10999]<br />
|-<br />
|[Du14] || '''O. Dudas''', [https://arxiv.org/abs/1011.5478 ''Coxeter orbits and Brauer trees II''], Int. Math. Res. Not. '''15''' (2014), 4100-4123.<br />
|-<br />
|[Dü04] || '''O. Düvel''', ''On Donovan's conjecture'', J. Algebra '''272''' (2004), 1-26.<br />
|- id="E"<br />
|[Ea16] || '''C. W. Eaton''', ''Morita equivalence classes of <math>2</math>-blocks of defect three'', Proc. AMS '''144''' (2016), 1961-1970.<br />
|-<br />
|[Ea18] || '''C. W. Eaton''', [https://arxiv.org/abs/1612.03485 ''Morita equivalence classes of blocks with elementary abelian defect groups of order 16''], [https://arxiv.org/abs/1612.03485 arXiv:1612.03485]<br />
|-<br />
|[Ea24] || '''C. W. Eaton''', [https://arxiv.org/abs/2401.04028 ''Blocks whose defect groups are Suzuki 2-groups''], [https://arxiv.org/abs/2401.04028 arXiv:2401.04028]<br />
|-<br />
|[EEL18] || '''C. W. Eaton, F. Eisele and M. Livesey''', [https://arxiv.org/abs/1809.08152 ''Donovan's conjecture, blocks with abelian defect groups and discrete valuation rings''], Math. Z. '''295''' (2020), 249-264.<br />
|-<br />
|[EKKS14] || '''C. W. Eaton, R. Kessar, B. Külshammer and B. Sambale''', ''<math>2</math>-blocks with abelian defect groups'', Adv. Math. '''254''' (2014), 706-735.<br />
|-<br />
|[EKS12] || '''C. W. Eaton, B. Külshammer and B. Sambale''', ''<math>2</math>-blocks with minimal nonabelian defect groups, II'', J. Group Theory '''15''' (2012), 311-321.<br />
|-<br />
|[EL18a] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/1709.04331 Classifying blocks with abelian defect groups of rank 3 for the prime 2]'', J. Algebra '''515''' (2018), 1-18.<br />
|-<br />
|[EL18b] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/1803.03539 Donovan's conjecture and blocks with abelian defect groups]'', Proc. AMS. '''147''' (2019), 963-970.<br />
|-<br />
|[EL18c] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/1810.10950 Some examples of Picard groups of blocks]'', J. Algebra '''558''' (2020), 350-370.<br />
|-<br />
|[EL20] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/2006.11173 Donovan's conjecture and extensions by the centralizer of a defect group]'', J. Algebra '''582''' (2021), 157-176.<br />
|-<br />
|[EL23] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/2310.05734 Morita equivalence classes of <math>2</math>-blocks with abelian defect groups of rank <math>4</math>]'', [https://arxiv.org/abs/2310.05734 arxiv:2310.05734]<br />
|-<br />
|[Ei16] || '''F. Eisele''', ''Blocks with a generalized quaternion defect group and three simple modules over a <math>2</math>-adic ring'', J. Algebra '''456''' (2016), 294-322.<br />
|-<br />
|[Ei18] || '''F. Eisele''', ''[https://arxiv.org/abs/1807.05110 The Picard group of an order and Külshammer reduction]'', Algebr. Represent. Theory '''24''' (2021), 505-518.<br />
|-<br />
|[Ei19] || '''F. Eisele''', ''[https://arxiv.org/abs/1908.00129 On the geometry of lattices and finiteness of Picard groups]'', J. Reine Angew. Math. '''782''' (2022), 219-333. <br />
|-<br />
|[EiLiv20] || '''F. Eisele and M. Livesey''', ''[https://arxiv.org/abs/2006.13837 Arbitrarily large Morita Frobenius numbers]'', Algebra Number Theory '''16''' (2022), 1889-1904.<br />
|-<br />
|[Er82] || '''K. Erdmann''', ''Blocks whose defect groups are Klein four groups: a correction'', J. Algebra '''76''' (1982), 505-518.<br />
|-<br />
|[Er87] || '''K. Erdmann''', ''Algebras and dihedral defect groups'', Proc. LMS '''54''' (1987), 88-114.<br />
|-<br />
|[Er88a] || '''K. Erdmann''', ''Algebras and quaternion defect groups, I'', Math. Ann. '''281''' (1988), 545-560.<br />
|-<br />
|[Er88b] || '''K. Erdmann''', ''Algebras and quaternion defect groups, II'', Math. Ann. '''281''' (1988), 561-582. <br />
|-<br />
|[Er88c] || '''K. Erdmann''', ''Algebras and semidihedral defect groups I'', Proc. LMS '''57''' (1988), 109-150. <br />
|-<br />
|[Er90] || ''' K. Erdmann''', ''Blocks of tame representation type and related algebras'', Lecture Notes in Mathematics '''1428''', Springer-Verlag (1990).<br />
|-<br />
|[Er90b] || '''K. Erdmann''', ''Algebras and semidihedral defect groups II'', Proc. LMS '''60''' (1990), 123-165.<br />
|- id="F"<br />
|[Fa17] || '''N. Farrell''', ''On the Morita Frobenius numbers of blocks of finite reductive groups'', J. Algebra '''471''' (2017), 299-318.<br />
|-<br />
|[FK18] || '''N. Farrell and R. Kessar''', [https://arxiv.org/abs/1805.02015 ''Rationality of blocks of quasi-simple finite groups''], Represent. Theory '''23''' (2019), 325-349.<br />
|- id="G"<br />
|[GMdelR21] || '''D. Garcia, l. Margolis and A. del Rio''', [https://arxiv.org/abs/2016.07231 ''Non-isomorphic 2-groups with isomorphic modular group algebras''], J. Reine Angew. Math. '''f783''' (2022), 269–274.<br />
|-<br />
|[GO97] || '''H. Gollan and T. Okuyama''', ''Derived equivalences for the smallest Janko group'', preprint (1997).<br />
|-<br />
|[GT19] || '''R. M. Guralnick and Pham Huu Tiep''', ''Sectional rank and Cohomology'', J. Algebra (2019) https://doi.org/10.1016/j.jalgebra.2019.04.023<br />
|- id="H"<br />
|[HM07] || '''G. T. Helleloid and U. Martin''', ''The automorphism group of a finite <math>p</math>-group is almost always a <math>p</math>-group'', J. Algebra (2007), 294-329.<br />
|-<br />
|[HP94] || '''H-W. Henn and S. Priddy''', ''<math>p</math>-nilpotence, classifying space indecompsability, and other properties of almost finite groups'', Comment. Math. Helvetici (1994), 335-350.<br />
|-<br />
|[Hi63] || '''G. Higman''', ''Suzuki 2-groups'', Illinois J. Math. '''7''' (1963), 79–96.<br />
|- <br />
|[HK00] || '''G. Hiss and R. Kessar''', ''Scopes reduction and Morita equivalence classes of blocks in finite classical groups'', J. Algebra '''230''' (2000), 378-423.<br />
|-<br />
|[HK05] || '''G. Hiss and R. Kessar''', ''Scopes reduction and Morita equivalence classes of blocks in finite classical groups II'', J. Algebra '''283''' (2005), 522-563.<br />
|-<br />
|[Ho97] || '''T. Holm''', ''Derived equivalent tame blocks'', J. Algebra '''194''' (1997), 178-200.<br />
|-<br />
|[HKL07] || '''T. Holm, R. Kessar and M. Linckelmann''', ''Blocks with a quaternion defect group over a 2-adic ring: the case <math>\tilde{A}_4</math>'', Glasgow Math. J. '''49''' (2007), 29–43.<br />
|- id="J"<br />
|[Ja69] || '''G. Janusz''', ''Indecomposable modules for finite groups'', Ann. Math. '''89''' (1969), 209-241.<br />
|-<br />
|[Jo96] || '''T. Jost''', ''Morita equivalences for blocks of finite general linear groups'', Manuscripta Math. '''91''' (1996), 121-144.<br />
|- id="K"<br />
|[Ke96] || '''R. Kessar''', ''Blocks and source algebras for the double covers of the symmetric and alternating groups'', J. Algebra '''186''' (1996), 872-933.<br />
|-<br />
|[Ke00] || '''R. Kessar''', ''Equivalences for blocks of the Weyl groups'', Proc. Amer. Math. Soc. '''128''' (2000), 337-346.<br />
|-<br />
|[Ke01] || '''R. Kessar''', ''Source algebra equivalences for blocks of finite general linear groups over a fixed field'', Manuscripta Math. '''104''' (2001), 145-162. <br />
|-<br />
|[Ke02] || '''R. Kessar''', ''Scopes reduction for blocks of finite alternating groups'', Quart. J. Math. '''53''' (2002), 443-454.<br />
|-<br />
|[Ke05] || ''' R. Kessar''', ''A remark on Donovan's conjecture'', Arch. Math (Basel) '''82''' (2005), 391-394.<br />
|-<br />
|[KL18] || '''R. Kessar and M. Linckelmann''', [https://arxiv.org/abs/1705.07227 ''Descent of equivalences and character bijections''], [https://arxiv.org/abs/1705.07227 arXiv:1705.07227]<br />
|-<br />
|[Ki84] || '''M. Kiyota''', ''On 3-blocks with an elementary abelian defect group of order 9'', J. Fac. Sci. Univ. Tokyo Sect. IA Math. '''31''' (1984), 33–58.<br />
|-<br />
|[Ko03] || '''S. Koshitani''', ''Conjectures of Donovan and Puig for principal <math>3</math>-blocks with abelian defect groups'', Comm. Alg. '''31''' (2003), 2229-2243; ''Corrigendum'', '''32''' (2004), 391-393.<br />
|-<br />
|[KKW02] || '''S. Koshitani, N. Kunugi and K. Waki''', ''Broué's conjecture for non-principal 3-blocks of finite groups'', J. Pure and Applied Algebra '''173''' (2002), 177-211. <br />
|-<br />
|[KKW04] || '''S. Koshitani, N. Kunugi and K. Waki''', ''Broué's abelian defect group conjecture for Held group and the sporadic Suzuki group'', J. Algebra '''279''' (2004), 638-666. <br />
|-<br />
|[KoLa20] || '''S. Koshitani and C. Lassueur''', ''Splendid Morita equivalences for principal 2-blocks with dihedral defect groups'', Math. Z. '''294''' (2020), 639-666.<br />
|-<br />
|[KoLa20b] || '''S. Koshitani and C. Lassueur''', ''Splendid Morita equivalences for principal blocks with generalised quaternion defect groups'', J. Algebra '''558''' (2020), 523-533.<br />
|-<br />
|[KoLaSa22] || '''S. Koshitani, C. Lassueur and B. Sambale''', ''Splendid Morita equivalences for principal blocks with semidihedral defect groups'', Proceedings of the American Mathematical Society '''150''' (2022), 41-53.<br />
|-<br />
|[KoLaSa23] || '''S. Koshitani, C. Lassueur and B. Sambale''', [https://arxiv.org/abs/2310.13621 ''Principal <math>2</math>-blocks with wreathed defect groups up to splendid Morita equivalence''], [https://arxiv.org/abs/2310.13621 arxiv:2310.13621]<br />
|-<br />
|[Kü80] || '''B. Külshammer''', ''On 2-blocks with wreathed defect groups'', J. Algebra '''64''' (1980), 529–555.<br />
|-<br />
|[Kü81] || '''B. Külshammer''', ''On p-blocks of p-solvable groups'', Comm. Alg. '''9''' (1981), 1763-1785.<br />
|-<br />
|[Kü91] || '''B. Külshammer''', ''Group-theoretical descriptions of ring-theoretical invariants of group algebras'', in Representation Theory of Finite Groups and Finite-Dimensional Algebras (Bielefeld, 1991), Progr. Math. '''95''', pp. 425-442, Birkhauser (1991).<br />
|-<br />
|[Kü95] || '''B. Külshammer''', ''Donovan's conjecture, crossed products and algebraic group actions'', Israel J. Math. '''92''' (1995), 295-306.<br />
|-<br />
|[KS13] || '''B. Külshammer and B. Sambale''', ''The 2-blocks of defect 4'', Representation Theory '''17''' (2013), 226-236.<br />
|-<br />
|[Ku00] || '''N. Kunugi''', ''Morita equivalent 3-blocks of the 3-dimensional projective special linear groups'', Proc. LMS '''80''' (2000), 575-589.<br />
|-<br />
|[Kup69] || '''H. Kupisch''', ''Unzerlegbare Moduln endlicher Gruppen mit zyklischer p-Sylow Gruppe'', Math. Z. '''108''' (1969), 77-104.<br />
|- id="L"<br />
|[LM80]||'''P. Landrock and G. O. Michler''', ''Principal 2-blocks of the simple groups of Ree type'', Trans. AMS '''260''' (1980), 83-111.<br />
|-<br />
|[Li94] || '''M. Linckelmann''', ''The source algebras of blocks with a Klein four defect group'', J. Algebra '''167''' (1994), 821-854.<br />
|-<br />
|[Li94b] || '''M. Linckelmann''', ''A derived equivalence for blocks with dihedral defect groups'', J. Algebra '''164''' (1994), 244-255. <br />
|-<br />
|[Li96] || '''M. Linckelmann''', ''The isomorphism problem for cyclic blocks and their source algebras'', Invent. Math. '''125''' (1996), 265-283.<br />
|-<br />
|[Li18] || '''M. Linckelmann''', [https://arxiv.org/abs/1805.08884 ''The strong Frobenius numbers for cyclic defect blocks are equal to one''], [https://arxiv.org/abs/1805.08884 arXiv:1805.08884]<br />
|-<br />
|[Li18b] || '''M. Linckelmann''', ''Finite-dimensional algebras arising as blocks of finite group algebras'', Contemporary Mathematics '''705''' (2018), 155-188.<br />
|-<br />
|[Li18c] || '''M. Linckelmann''', ''The block theory of finite group algebras, Volume 1'', London Math. Soc. Student Texts '''92''', Cambridge University Press (2018).<br />
|-<br />
|[Li18d] || '''M. Linckelmann''', ''The block theory of finite group algebras, Volume 2'', London Math. Soc. Student Texts '''92''', Cambridge University Press (2018).<br />
|-<br />
|[LM20] || '''M. Linckelmann and W. Murphy''', [https://arxiv.org/abs/2005.02223 ''A 9-dimensional algebra which is not a block of a finite group''], Quarterly Journal of Mathematics 72 (2021), 1077–1088<br />
|-<br />
|[Liv19] || '''M. Livesey''', [https://arxiv.org/abs/1907.12167 ''On Picard groups of blocks with normal defect groups''], J. Algebra '''566''' (2021), 94-118.<br />
|-<br />
|[LiMa20] || '''M. Livesey and C. Marchi''', [https://arxiv.org/abs/2002.10571 ''On Picent for blocks with normal defect group''], [https://arxiv.org/abs/2002.10571 arXiv:2002.10571]<br />
|-<br />
|[LiMa20b] || '''M. Livesey and C. Marchi''', [https://arxiv.org/abs/2008.05857 ''Picard groups for blocks with normal defect groups and linear source bimodules''], [https://arxiv.org/abs/2008.05857 arXiv:2008.05857]<br />
|- id="M"<br />
|[Mac] || '''N. Macgregor''', ''Morita equivalence classes of tame blocks of finite groups'', J. Algebra '''608''' (2022), 719-754.<br />
|-<br />
|[Mar] || '''C. Marchi''', ''Picard groups for blocks'', PhD thesis, University of Manchester (2022)<br />
|-<br />
|[Ma86] || '''U. Martin''', ''Almost all <math>p</math>-groups have automorphism group a <math>p</math>-group'', Bull. AMS '''15''' (1986), 78-82.<br />
|-<br />
|[McK19] || '''E. McKernon''', [https://arxiv.org/abs/1912.03222 ''2-Blocks whose defect group is homocyclic and whose inertial quotient contains a Singer cycle''], J. Algebra '''563''' (2020), 30–48.<br />
|-<br />
|[MS08] || '''J. Müller and M. Schaps''', ''The Broué conjecture for the faithful 3-blocks of <math>4.M_{22}</math>'', J. Algebra '''319''' (2008), 3588-3602.<br />
|- id="N"<br />
|[NS18] || '''G. Navarro and B. Sambale''', ''On the blockwise modular isomorphism problem'', Manuscripta Math. '''157''' (2018), 263-278.<br />
|- <br />
|[Ne02] || '''G. Nebe''', [http://www.math.rwth-aachen.de/~Gabriele.Nebe/papers/survey.pdf ''Group rings of finite groups over p-adic integers, some examples''], Proceedings of the conference Around Group rings (Edmonton) Resenhas '''5''' (2002), 329-350.<br />
|- id="O"<br />
|[Ok97] || '''T. Okuyama''', ''Some examples of derived equivalent blocks of finite groups'', preprint (1997).<br />
|- id="P"<br />
|[Pu88]|| '''L. Puig''', ''Nilpotent blocks and their source algebras'', Invent. Math. '''93''' (1988), 77-116.<br />
|-<br />
|[Pu94] || '''L. Puig''', ''On Joanna Scopes’ criterion of equivalence for blocks of symmetric groups'', Algebra Colloq. '''1''' (1994), 25-55.<br />
|-<br />
|[Pu99]|| '''L. Puig''', ''On the local structure of Morita and Rickard equivalences between Brauer blocks'', Progress in Math. '''178''', Birkhauser Verlag (1999).<br />
|-<br />
|[Pu09] || '''L. Puig''', ''Block source algebras in p-solvable groups'', Michigan Math. J. '''58''' (2009), 323-338.<br />
|- id="R"<br />
|[Ri96] || '''J. Rickard''', ''Splendid equivalences: derived categories and permutation modules'', Proc. London Math. Soc. '''72''' (1996), 331-358.<br />
|-<br />
|[Ro95] || '''R. Rouquier''', ''From stable equivalences to Rickard equivalences for blocks with cyclic defect'', Proceedings of Groups 1993, Galway-St. Andrews Conference, Vol. 2, London Math. Soc. Lecture Note Ser. '''212''', Cambridge University Press (1995), 512-523.<br />
|-<br />
|[Ru11] || '''P. Ruengrot''', ''Perfect isometry groups for blocks of finite groups'', PhD Thesis, University of Manchester (2011).<br />
|- id="S"<br />
|[Sa11] || '''B. Sambale''', ''<math>2</math>-blocks with minimal nonabelian defect groups'', J. Algebra '''337''' (2011), 261–284.<br />
|-<br />
|[Sa12] || '''B. Sambale''', ''Blocks with defect group <math>D_{2^n} \times C_{2^m}</math>'', J. Pure Appl. Algebra '''216''' (2012), 119–125.<br />
|-<br />
|[Sa12b] || '''B. Sambale''', ''Fusion systems on metacyclic 2-groups'', Osaka J. Math. '''49''' (2012), 325–329.<br />
|-<br />
|[Sa13] || '''B. Sambale''', ''Blocks with defect group <math>Q_{2^n} \times C_{2^m}</math> and <math>SD_{2^n} \times C_{2^m}</math>'', Algebr. Represent. Theory '''16''' (2013), 1717–1732.<br />
|-<br />
|[Sa13b] || '''B. Sambale''', ''Blocks with central product defect group <math>D_{2^n} ∗ C_{2^m}</math>'', Proc. Amer. Math. Soc. '''141''' (2013), 4057–4069.<br />
|-<br />
|[Sa13c] || '''B. Sambale''', ''Further evidence for conjectures in block theory'', Algebra Number Theory '''7''' (2013), 2241–2273. <br />
|-<br />
|[Sa14] || '''B. Sambale''', ''Blocks of Finite Groups and Their Invariants'', Lecture Notes in Mathematics, Springer (2014).<br />
|-<br />
|[Sa16] || '''B. Sambale''', ''2-blocks with minimal nonabelian defect groups III'', Pacific J. Math. '''280''' (2016), 475–487.<br />
|-<br />
|[Sa20] || '''B. Sambale''', [https://arxiv.org/abs/2005.13172 ''Blocks with small-dimensional basic algebra''], Bul. Aust. Math. Soc. '''103''' (2021), 461-474.<br />
|-<br />
|[SSS98] || '''M. Schaps, D. Shapira and O. Shlomo''', ''Quivers of blocks with normal defect groups'', Proc. Symp. in Pure Mathematics '''63''', Amer. Math. Soc. (1998), 497-510.<br />
|-<br />
|[Sc91] || '''J. Scopes''', ''Cartan matrices and Morita equivalence for blocks of the symmetric groups'', J. Algebra '''142''' (1991), 441-455.<br />
|-<br />
|[Sh20] || '''V. Shalotenko''', ''Bounds on the dimension of Ext for finite groups of Lie type'', J. Algebra '''550''' (2020), 266-289.<br />
|-<br />
|[St02] || '''R. Stancu''', ''Almost all generalized extraspecial p-groups are resistant'', J. Algebra '''249''' (2002), 120-126.<br />
|-<br />
|[St06] || '''R. Stancu''', ''Control of fusion in fusion systems'', J. Algebra and its Applications '''5''' (2006), 817-837. <br />
|- id="T"<br />
|[Th93] || '''J. Thévenaz''', ''Most finite groups are <math>p</math>-nilpotent'', Exposition. Math. '''11''' (1993), 359-363.<br />
|- id="V"<br />
|[vdW91] || '''R. van der Waall''', ''On p-nilpotent forcing groups'', Indag. Mathem., N.S., '''2''' (1991), 367-384.<br />
|- id="W"<br />
|[Wa00]|| '''A. Watanabe''', ''A remark on a splitting theorem for blocks with abelian defect groups'', RIMS Kokyuroku '''1140''', Edited by H.Sasaki, Research Institute for Mathematical Sciences, Kyoto University (2000), 76-79.<br />
|-<br />
|[WZZ18] || '''Chao Wu, Kun Zhang and Yuanyang Zhou''', ''[https://arxiv.org/abs/1803.07262 Blocks with defect group <math>Z_{2^n} \times Z_{2^n} \times Z_{2^m}</math>]'', J. Algebra '''510''' (2018), 469-498.<br />
|}</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Generic_classifications_by_p-group_class&diff=1235
Generic classifications by p-group class
2024-01-09T11:15:50Z
<p>Charles Eaton: Suzuki 2-groups</p>
<hr />
<div>This page contains results for classes of ''p''-groups for which we either have classifications or have general results concerning Morita equivalence classes. <br />
<br />
== Fusion trivial ''p''-groups ==<br />
<br />
''p''-groups <math>P</math> for which the only saturated fusion system is <math>\mathcal{F}_P(P)</math> have not yet been given a name in the literature (to our knowledge). We will call them ''fusion trivial'', but ''nilpotent forcing'' also seems appropriate following [[References#V|[vdW91]]] (where ''p''-groups for which any finite group <math>G</math> containing <math>P</math> as a Sylow ''p''-subgroup must be ''p''-nilpotent are called ''p-nilpotent forcing''). It is not known whether these two definitions are equivalent, i.e., whether there exist ''p''-nilpotent forcing ''p''-groups for which there is an exotic fusion system.<br />
<br />
Blocks with fusion trivial defect groups must be nilpotent and so Morita equivalent to the group algebra of a defect group by [[References#P|[Pu88]]].<br />
<br />
Examples of fusion trivial ''p''-groups are abelian ''2''-groups with automorphism group a ''2''-group (i.e., those whose cyclic factors have pairwise distinct orders), and metacyclic 2-groups other than homocyclic, dihedral, generalised quaternion or semidihedral groups (see [[References#C|[CG12]]] or [[References#S|[Sa12b]]]).<br />
<br />
Note that a ''p''-group is fusion trivial if and only if it is resistant and has automorphism group a ''p''-group. See [[References#S|[St06]]] for an analysis of resistant ''p''-groups.<br />
<br />
== Cyclic ''p''-groups ==<br />
<br />
[[Blocks with cyclic defect groups|Click here for background on blocks with cyclic defect groups]].<br />
<br />
Morita equivalence classes are labelled by [[Brauer trees]], but it is at present an open problem as to which Brauer trees are realised by blocks of finite groups. Each ''k''-Morita equivalence class corresponds to an unique <math>\mathcal{O}</math>-Morita equivalence class. <br />
<br />
For <math>p=2,3</math> every appropriate Brauer tree is realised by a block and we can give generic descriptions.<br />
<br />
[[C(2^n)|<math>2</math>-blocks with cyclic defect groups]]<br />
<br />
[[C(3^n)|<math>3</math>-blocks with cyclic defect groups]]<br />
<br />
== Tame blocks ==<br />
<br />
[[Tame blocks|Click here for background on tame blocks]].<br />
<br />
Erdmann classified algebras which are candidates for [[Basic algebras|basic algebras]] of tame blocks, i.e., those with dihedral, semidihedral or generalised quaternion defect groups (see [[References|[Er90] ]]) and in the cases of dihedral and semihedral defect groups determined which are realised by blocks of finite groups. In the case of generalised quaternion groups, the case of blocks with two simple modules is still open. These classifications only hold with respect to the field ''k'' at present.<br />
<br />
Principal blocks with dihedral defect groups are classified up to source algebra equivalence in [[References#K|[KoLa20]]].<br />
<br />
== Abelian <math>2</math>-groups with <math>2</math>-rank at most four ==<br />
<br />
[[Image:under-construction.png|50px|left]]<br />
<br />
These have been classified in [[References#W|[WZZ18]]], [[References#E|[EL18a]]] and [[References#E|[EL23]]] with respect to <math>\mathcal{O}</math>. The derived equivalences classes with respect to <math>\mathcal{O}</math> are known.<br />
<br />
Let <math>l,m,n,r \geq 2</math> be distinct.<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>rk_2(D) \leq 4</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>D</math> <br />
! scope="col"| # <math>(\mathcal{O}</math>-)Morita classes<br />
! scope="col"| # <math>\mathcal{O}</math>-Derived equiv classes<br />
! scope="col"| Always endopermutation source Morita equivalent?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|[[C(2^n)|<math>C_{2^n}</math>]] || 1 || 1 || Yes || [[References#L|[Li96]]] || [[Blocks with cyclic defect groups]]<br />
|-<br />
|[[C2xC2|<math>C_2 \times C_2</math>]] || 3 || 2 || Yes || [[References#L|[Li94]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)|<math>C_{2^m} \times C_{2^m}</math>]] || 2 || 2 || Yes || [[References#E|[EKKS14]]] ||<br />
|-<br />
|[[C(2^m)xC(2^n)|<math>C_{2^m} \times C_{2^n}</math>]] || 1 || 1 || Yes || [[References#P|[Pu88]]] || [[Nilpotent blocks]]<br />
|-<br />
|[[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]] || 8 || 4 || || [[References#E|[Ea16]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)xC(2^m)|<math>C_{2^m} \times C_{2^m} \times C_{2^m}</math>]] || 4 || 4 || Yes || [[References#W|[WZZ18]]] ||<br />
|-<br />
|[[C(2^m)xC2xC2|<math>C_{2^m} \times C_2 \times C_2</math>]] || 3 || 2 || || [[References#E|[EL18a]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)xC(2^n)|<math>C_{2^m} \times C_{2^m} \times C_{2^n}</math>]] || 2 || 2 || || [[References#W|[WZZ18]]] ||<br />
|-<br />
|[[C(2^l)xC(2^m)xC(2^n)|<math>C_{2^l} \times C_{2^m} \times C_{2^n}</math>]] || 1 || 1 || Yes || [[References#P|[Pu88]]] || [[Nilpotent blocks]]<br />
|-<br />
|[[(C2)^4|<math>(C_2)^4</math>]] || 16 || || || [[References#E|[Ea18]]] ||<br />
|-<br />
|[[C(2^m)xC2xC2xC2|<math>C_{2^m} \times C_2 \times C_2 \times C_2</math>]] || || || || [[References#E|[EL23]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)xC2xC2|<math>C_{2^m} \times C_{2^m} \times C_2 \times C_2</math>]] || || || || [[References#E|[EL23]]] ||<br />
|-<br />
|[[C(2^m)xC(2^n)xC2xC2|<math>C_{2^m} \times C_{2^n} \times C_2 \times C_2</math>]] || || || || [[References#E|[EL23]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)xC(2^n)xC(2^n)|<math>C_{2^m} \times C_{2^m} \times C_{2^n} \times C_{2^n}</math>]] || || || || [[References#E|[EL23]]] ||<br />
|-<br />
|[[C(2^l)xC(2^l)xC(2^m)xC(2^n)|<math>C_{2^l} \times C_{2^l} \times C_{2^m} \times C_{2^n}</math>]] || || || || [[References#E|[EL23]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)xC(2^m)xC(2^n)|<math>C_{2^m} \times C_{2^m} \times C_{2^m} \times C_{2^n}</math>]] || || || || [[References#E|[EL23]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)xC(2^m)xC(2^m)|<math>C_{2^m} \times C_{2^m} \times C_{2^m} \times C_{2^m}</math>]] || || || || [[References#E|[EL23]]] ||<br />
|-<br />
|[[C(2^l)xC(2^m)xC(2^n)xC(2^r)|<math>C_{2^l} \times C_{2^m} \times C_{2^n} \times C_{2^n}</math>]] || || || || [[References#E|[EL23]]] || [[Nilpotent blocks]] <br />
|}<br />
<br />
<br />
<!--== Abelian <math>2</math>-groups ==<br />
<br />
Donovan's conjecture holds for ''2''-blocks with abelian defect groups. Some generic classification results are known for certain inertial quotients. These will be detailed here.--><br />
<br />
== Minimal nonabelian <math>2</math>-groups ==<br />
<br />
Blocks with defect groups which are minimal nonabelian <math>2</math>-groups of the form <math>P=\langle x,y:x^{2^r}=y^{2^r}=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle</math> are classified in [[References#E|[EKS12]]]. There are two <math>\mathcal{O}</math>-Morita equivalence classes, with representatives <math>\mathcal{O}P</math> and <math>\mathcal{O}(P:C_3)</math>.<br />
<br />
For arbitrary minimal nonabelian <math>2</math>-groups, by [[References#S|[Sa16]]] blocks with such defect groups and the same fusion system are isotypic.<br />
<br />
== Homocyclic <math>2</math>-groups when inertial quotient contains a Singer cycle == <br />
<br />
A Singer cycle is an element of order <math>p^n-1</math> in <math>\operatorname{Aut}({C_p}^n) \cong GL_n(p)</math>, and a subgroup generated by a Singer cycle acts transitively on the non-trivial elements of <math>(C_p)^n</math>.<br />
<br />
<math>2</math>-blocks with homocyclic defect group <math> D \cong (C_{2^m})^n </math> whose inertial quotient <math> \mathbb{E} </math> contains a Singer cycle are classified in [[References#E|[McK19]]]. In this situation, <math>\mathbb{E}</math> has the form <math>E:F </math> where <math>E \cong C_{2^n-1}</math> and <math>F</math> is trivial or a subgroup of <math>C_n</math>. There are three <math> \mathcal{O} </math>-Morita equivalence classes when <math>m=1, n =3 </math>; two when <math>m=1 , n \neq 3 </math>; and only one when <math>m> 1 </math>. The three classes have representatives [[M(8,5,7) | <math> \mathcal{O}J_1 </math>]], which occurs only when <math>m=1, n=3</math>; <math> \mathcal{O}(SL_2(2^n):F) </math>, which occurs only when <math>m=1 </math>; and <math> \mathcal{O}(D : \mathbb{E})</math>. <br />
<br />
The Morita equivalence between the block and the class representative is known to be basic, possibly except when <math>m=1,n=3</math>, since the Morita equivalences between the [[M(8,5,8) |principal block of <math>{\rm \operatorname{Aut}(SL_2(8))}</math>]] and the blocks [[M(8,5,8)#Other_notatable_representatives |<math>B_0(\mathcal{O}({}^2G_2(3^{2m+1})))</math>]] are not known to be basic.<br />
<br />
<br />
== Abelian <math>2</math>-groups when inertial quotient is cyclic and acts freely on the defect group == <br />
<br />
<math>2</math>-blocks with abelian defect group <math> D </math> whose inertial quotient <math> E </math> is a cyclic group that acts freely on the defect group (i.e. such that the stabiliser in <math> E </math> of any nontrivial element of <math>D</math> is trivial) are classified in [[References#E|[ArMcK20]]].<br />
<br />
If the action of <math>E</math> is transitive, then <math>D</math> is homocyclic, <math>E</math> is a Singer cycle and, by the previous section, the block is either Morita equivalent to the principal block of <math> \mathcal{O}SL_2(2^n) </math>, or to <math> \mathcal{O}(D : E)</math>. <br />
<br />
If the action of <math>E</math> is not transitive, then the block is Morita equivalent to <math> \mathcal{O}(D : E)</math>. <br />
<br />
In each case, the Morita equivalence between the block and the class representative is known to be basic.<br />
<br />
== Extraspecial <math>p</math>-groups of order <math>p^3</math> and exponent <math>p</math> for <math>p \geq 5</math> ==<br />
<br />
These blocks are described in [[References#A|[AE23]]]. Donovan's conjecture holds in this case.<br />
<br />
== Principal blocks with wreath products <math>C_{2^n} \wr C_2</math> defect groups ==<br />
<br />
These are classified up to splendid Morita equivalence in [[References#K|[KoLaSa23]]]. There are six classes for each <math>n</math>, with representatives the principal blocks of:<br />
<br />
<math>C_{2^n} \wr C_2</math><br />
<br />
<math>(C_{2^n} \times C_{2^n}):S_3</math><br />
<br />
<math>SL_2^n(q)</math> where <math>(q-1)_2=2^n</math>, consisting of matrices whose determinant is a <math>2^n</math> root of unity<br />
<br />
<math>SU_2^n(q)</math> where <math>(q+1)_2=2^n</math>, consisting of matrices whose determinant is a <math>2^n</math> root of unity <br />
<br />
<math>PSL_3(q)</math> where <math>(q-1)_2=2^n</math><br />
<br />
<math>PSU_3(q)</math> where <math>(q+1)_2=2^n</math><br />
<br />
== Blocks whose defect groups are Suzuki <math>2</math>-groups ==<br />
<br />
These blocks are described in [[References#E|[Ea24]]]. Donovan's conjecture holds in this case. Suzuki <math>2</math>-groups are non-abelian <math>2</math>-groups <math>P</math> with more than one involution for which there is <math>\varphi \in {\rm Aut}(P)</math> permuting the involutions in <math>P</math> transitively. By [[References#|[Hi63]]] these satisfy <math>\Omega_1(P)=Z(P)=\Phi(P)=[P,P]</math> and fall into classes A, B, C and D. The Sylow <math>2</math>-subgroups of the Suzuki simple groups and of <math>PSU_3(2^n)</math> fall in classes A and B respectively. Morita equivalence classes have representatives as follows:<br />
<br />
a block of a finite group with a normal defect group<br />
<br />
the principal block of <math>H</math> for <math>^2B_2(2^{2n+1}) \leq H \leq {\rm Aut}({}^2B_2(2^{2n+1}))</math> for some <math>n \geq 1</math><br />
<br />
a block of maximal defect of <math>H</math> where <math>Z(H) \leq [H,H]</math> and <math>PSU_3(2^n) \leq H/Z(H) \leq {\rm Aut}(PSU_3(2^n))</math> for some <math>n \geq 2</math> with <math>[H/Z(H):PSU_3(2^n)]</math> odd</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Blocks_with_basic_algebras_of_low_dimension&diff=1234
Blocks with basic algebras of low dimension
2023-12-13T13:33:08Z
<p>Charles Eaton: Removed dim 9 non-example description</p>
<hr />
<div>== Blocks with basic algebras of dimension at most 16 ==<br />
<br />
In [[References#L|[Li18b]]] Markus Linckelmann calculated the <math>k</math>-algebras of dimension at most twelve which occur as basic algebras of blocks of finite groups, with the exception of one case of dimension 9 where no block with that basic algebra was identified. This final case was ruled out by Linckelmann and Murphy in [[References#L|[LM20]]]. Using the classification of finite simple groups, the basic algebras of dimension 13 or 14 for blocks of finite groups were calculated by Sambale in [[References#S|[Sa20]]]. Later Benson and Sambale in [[References#B|[BS23]]] gave a classification for dimensions 15 and 16, except for one unsettled case of a block with defect group <math>C_{13}</math> in dimension 15.<br />
<br />
The results are incorporated into the table below. <br />
<br />
{| class="wikitable"<br />
|-<br />
! scope="col"| Dimension<br />
! scope="col"| Class<br />
! scope="col"| Defect group<br />
! scope="col"| Representative<br />
! scope="col"| <math>\dim_k(Z(A))</math><br />
! scope="col"| <math>l(A)</math><br />
! scope="col"| Notes<br />
|-<br />
| 1 || [[M(1,1,1)]] || <math>1</math> || <math>k1</math> || 1 || 1 || Blocks of defect zero<br />
|-<br />
| 2 || [[M(2,1,1)]] || <math>C_2</math> || <math>kC_2</math> || 2 || 1 || <br />
|-<br />
| 3 || [[M(3,1,1)]] || <math>C_3</math> || <math>kC_3</math> || 3 || 1 || <br />
|-<br />
| 4 || [[M(4,1,1)]] || <math>C_4</math> || <math>kC_4</math> || 4 || 1 ||<br />
|-<br />
| 4 || [[M(4,2,1)]] || <math>C_2 \times C_2</math> || <math>k(C_2 \times C_2)</math> || 4 || 1 ||<br />
|-<br />
| 5 || [[M(5,1,1)]] || <math>C_5</math> || <math>kC_5</math> || 5 || 1 ||<br />
|-<br />
| 6 || [[M(3,1,2)]] || <math>C_3</math> || <math>kS_3</math> || 3 || 2 ||<br />
|-<br />
| 7 || [[M(5,1,3)]] || <math>C_5</math> || <math>B_0(kA_5)</math> || 4 || 2 ||<br />
|-<br />
| 7 || [[M(7,1,1)]] || <math>C_7</math> || <math>kC_7</math> || 7 || 1 ||<br />
|-<br />
| 8 || [[M(8,1,1)]] || <math>C_8</math> || <math>kC_8</math> || 8 || 1 ||<br />
|-<br />
| 8 || [[M(8,2,1)]] || <math>C_4 \times C_2</math> || <math>k(C_4 \times C_2)</math> || 8 || 1 ||<br />
|-<br />
| 8 || [[M(8,3,1)]] || <math>D_8</math> || <math>kD_8</math> || 5 || 1 ||<br />
|-<br />
| 8 || [[M(8,4,1)]] || <math>Q_8</math> || <math>kQ_8</math> || 5 || 1 ||<br />
|-<br />
| 8 || [[M(8,5,1)]] || <math>C_2 \times C_2 \times C_2</math> || <math>k(C_2 \times C_2 \times C_2)</math> || 8 || 1 ||<br />
|-<br />
| 8 || [[M(7,1,3)]] || <math>C_7</math> || <math>B_0(kPSL_2(13))</math> || 5 || 2 ||<br />
|-<br />
| 9 || [[M(9,1,1)]] || <math>C_9</math> || <math>kC_9</math> || 9 || 1 ||<br />
|-<br />
| 9 || [[M(9,1,3)]] || <math>C_9</math> || <math>B_0(kSL_2(8))</math> || 6 || 2 ||<br />
|-<br />
| 9 || [[M(9,2,1)]] || <math>C_3 \times C_3</math> || <math>k(C_3 \times C_3)</math> || 9 || 1 ||<br />
|-<br />
| 9 || [[M(9,2,23)]] || <math>C_3 \times C_3</math> || Faithful block of <math>k((C_3 \times C_3):Q_8)</math>, in which <math>Z(Q_8)</math> acts trivially || 6 || 1 || SmallGroup(72,24)<br />
|-<br />
| 10 || [[M(5,1,2)]] || <math>C_5</math> || <math>kD_{10}</math> || 4 || 2 ||<br />
|-<br />
| 10 || [[M(11,1,3)]] || <math>C_{11}</math> || <math>B_0(kSL_2(32))</math> || 7 || 2 ||<br />
|-<br />
| 11 || [[M(8,3,3)]] || <math>D_8</math> || <math>kS_4</math> || 5 || 2 ||<br />
|-<br />
| 11 || [[M(7,1,6)]] || <math>C_7</math> || <math>B_0(kA_7)</math> || 5 || 3 ||<br />
|-<br />
| 11 || [[M(11,1,1)]] || <math>C_{11}</math> || <math>kC_{11}</math> || 11 || 1 ||<br />
|-<br />
| 11 || [[M(13,1,3)]] || <math>C_{13}</math> || <math>B_0(kPSL_2(25))</math> || 8 || 2 ||<br />
|-<br />
| 12 || [[M(4,2,3)]] || <math>C_2 \times C_2</math> || <math>kA_4</math> || 4 || 3 ||<br />
|-<br />
| 13 || [[M(16,7,3)]] || <math>D_{16}</math> || <math>B_0(kPGL_2(7))</math> || 7 || 2 ||<br />
|-<br />
| 13 || [[M(16,8,4)]] || <math>SD_{16}</math> || <math>B_3(k(3.M_{10}))</math> || 7 || 2 ||<br />
|-<br />
| 13 || [[M(7,1,7)]] || <math>C_7</math> || <math>B_{15}(k6.A_7)</math> || 5 || 3 ||<br />
|-<br />
| 13 || [[M(13,1,1)]] || <math>C_{13}</math> || <math>kC_{13}</math> || 13 || 1 ||<br />
|-<br />
| 13 || M(13,1,?) || <math>C_{13}</math> || <math>B_0(kPSL_3(3))</math> || 7 || 3 ||<br />
|-<br />
| 13 || M(17,1,?) || <math>C_{17}</math> || <math>B_0(kPSL_2(16))</math> || 10 || 2 ||<br />
|-<br />
| 14 || [[M(5,1,5)]] || <math>C_5</math> || <math>B_0(kS_5)</math> || 5 || 4 ||<br />
|-<br />
| 14 || [[M(7,1,2)]] || <math>C_7</math> || <math>kD_{14}</math> || 5 || 2 ||<br />
|-<br />
| 14 || [[M(7,1,5)]] || <math>C_7</math> || <math>B_0(kPSL_3(3))</math> || 5 || 3 ||<br />
|-<br />
| 14 || M(19,1,?) || <math>C_{19}</math> || <math>B_0(kPSL_2(37))</math> || 11 || 2 ||<br />
|-<br />
| 15 || M(19,1,?) || <math>C_{19}</math> || <math>B_0(kGL_3(7))</math> || ||<br />
|-<br />
| 15 || || <math>C_{13}</math> || ?? || || ||<br />
<br />
|}</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Status_of_Donovan%27s_conjecture&diff=1233
Status of Donovan's conjecture
2023-10-24T09:06:21Z
<p>Charles Eaton: /* Donovan's conjecture by p-group */ Extraspecial and wreathed</p>
<hr />
<div>[[Image:Donovan.jpg|150px|thumb|right|Peter Donovan]]<br />
<br />
== Donovan's conjecture by <math>p</math>-group ==<br />
<br />
In the following, the column headed [[Statements of conjectures #Donovan's conjecture|Donovan's conjecture]] indicates whether the conjecture is known over <math>k</math> or <math>\mathcal{O}</math>.<br />
<br />
{| class="wikitable"<br />
|-<br />
! scope="col"| <math>p</math>-groups<br />
! scope="col"| Donovan's conjecture<br />
! scope="col"| Puig's conjecture<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|-<br />
|Cyclic <math>p</math>-groups || <math>\mathcal{O}</math> || Yes || [[References#L|[Li96]]] ||<br />
|-<br />
|<math>C_2 \times C_2</math> || <math>\mathcal{O}</math> || Yes || [[References#C|[CEKL11]]] || Donovan's conjecture without CFSG, Puig using CFSG<br />
|-<br />
|Abelian <math>2</math>-groups || <math>\mathcal{O}</math> || No || [[References#E|[EEL18]]] ||<br />
|-<br />
|Abelian <math>3</math>-groups || No || No || [[References#K|[Ko03]]] || Puig's conjecture known for principal blocks<br />
|-<br />
|Dihedral <math>2</math>-groups || <math>k</math> || No || [[References#E|[Er87]]] ||<br />
|-<br />
|Semidihedral <math>2</math>-groups || <math>k</math> || No || [[References#E|[Er88c], [Er90b]]] ||<br />
|-<br />
|<math>Q_8</math> || <math>\mathcal{O}</math> || No || [[References#E|[Er88a]]], [[References#E|[Er88b]]], [[References#K|[HKL07]]], [[References#E|[Ei16]]] ||<br />
|-<br />
|<math>Q_8 \times C_{2^n}</math> || <math>\mathcal{O}</math> || No || [[References#E|[EL20]]] ||<br />
|-<br />
|<math>Q_8 \times Q_8</math> || <math>\mathcal{O}</math> || No || [[References#E|[EL20]]] ||<br />
|-<br />
|Generalised quaternion <math>2</math>-groups || No || No || [[References#E|[Er88a], [Er88b]]] || Donovan's conjecture over <math>\mathcal{O}</math> known if <math>l(B) \neq 2</math><ref>When <math>l(B) \neq 2</math>, each <math>k</math>-Morita equivalence class lifts uniquely to <math>\mathcal{O}</math> by [[References|[Ei16]]].</ref><br />
|- <br />
|Minimal nonabelian <math>2</math>-groups of the form <math>\langle x,y:x^{2^r}=y^{2^r}=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle</math> || <math>\mathcal{O}</math> || No || [[References#E|[EKS12]]] || <br />
|-<br />
|Metacyclic noncyclic <math>2</math>-groups of nonmaximal class || <math>\mathcal{O}</math> || No || [[References#C|[CG12]]], [[References#S|[Sa12b]]] || All blocks nilpotent<br />
|-<br />
|<math>p_+^{1+2}</math> for <math>p \geq 5</math> || <math>\mathcal{O}</math> || No || [[References#A|[AE23]]] ||<br />
|-<br />
|<math>C_{2^n} \wr C_2</math> || Principal blocks (<math>\mathcal{O}</math>) || Principal blocks (<math>\mathcal{O}</math>) || [[References#K|[KoLaSa23]]] ||<br />
|}<br />
<br />
== Donovan's conjecture by class of group or block ==<br />
<br />
In the table, the column headed Donovan's conjecture indicates whether the conjecture is known over <math>k</math> or <math>\mathcal{O}</math>.<br />
<br />
Note that knowing the <math>\mathcal{O}</math>-Donovan conjecture or [[Statements of conjectures #Puig's conjecture|Puig's conjecture]] for blocks for a class of groups does not necessarily mean that the <math>\mathcal{O}</math>-lifts or source algebras of the <math>k</math>-Morita equivalence classes involved are known. This is only known for elements of the Morita equivalence class which occur as blocks of groups in that class.<br />
<br />
{| class="wikitable"<br />
|-<br />
! scope="col"| Groups<br />
! scope="col"| Blocks<br />
! scope="col"| Donovan's conjecture<br />
! scope="col"| Puig's conjecture<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|-<br />
|<math>p</math>-solvable groups || All || <math>\mathcal{O}</math> || Yes || Over <math>k</math> by [[References|[Ku81]]], Puig's conjecture by [[References|[Pu09]]] || See [[References#L|[Li18d,10.6.2]]]<br />
|-<br />
|Symmetric groups || All || <math>\mathcal{O}</math> || Yes || Over <math>k</math> by [[References#S|[Sc91]]], Puig's conjecture by [[References#P|[Pu94]]] ||<br />
|-<br />
|Double covers of symmetric groups || All || <math>\mathcal{O}</math> || Yes || [[References#K|[Ke96]]] ||<br />
|-<br />
|Alternating groups and their double covers || All || <math>\mathcal{O}</math> || Yes || [[References#K|[Ke02], [Ke96]]] ||<br />
|-<br />
|<math>GL_n(q)</math> for fixed <math>q</math> || Unipotent blocks || <math>\mathcal{O}</math> || Yes || Over <math>k</math> by [[References#J|[Jo96]]], Puig's conjecture by [[References#K|[Ke01]]] ||<br />
|-<br />
|Classical groups || Unipotent blocks for linear primes || <math>\mathcal{O}</math> || Yes || [[References#H|[HK00], [HK05]]] || Detailed results beyond those stated here<br />
|-<br />
|Weyl groups of type <math>B, D</math> || All || <math>\mathcal{O}</math> || Yes || [[References#K|[Ke00]]] ||<br />
|-<br />
|Arbitrary groups || Blocks with [[Glossary#Trivial intersection subgroup|trivial intersection]] defect groups || <math>\mathcal{O}</math> || No || [[References#A|[AE04]]] ||<br />
|}<br />
<br />
== Weak Donovan conjecture ==<br />
<br />
As described in [[References #D|[Dü04]]] the [[Statements of conjectures #Weak Donovans conjecture|Weak Donovan conjecture]] is equivalent to [[Statements of conjectures #Weak Donovans conjecture|bounding the dimensions of the Ext spaces between simple modules]] and [[Statements of conjectures #Weak Donovans conjecture|bounding the Loewy length]]. See [[References #G|[GT19]]] and [[References #S|[Sh20]]] for progress on the former problem.<br />
<br />
== Notes ==<br />
<br />
<references /></div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Classification_by_p-group&diff=1232
Classification by p-group
2023-10-24T08:58:13Z
<p>Charles Eaton: /* Blocks for p=2 */ SmallGroup(32,11), SmallGroup(32,45)</p>
<hr />
<div>'''Classification of Morita equivalences for blocks with a given defect group'''<br />
<br />
On this page we list classifications of Morita equivalence classes for each isomorphism class of p-groups in turn. [[Generic classifications by p-group class|Generic classifications for classes of p-groups can be found here]].<br />
<br />
See [[Labelling for Morita equivalence classes|this page]] for a description of the labelling conventions.<br />
<br />
== Blocks for <math> p=2 </math> ==<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>1 \leq |D| \leq 8</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|- <br />
| 2 || [[C2|1]] || [[C2|<math>C_2</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
| 4 || [[C4|1]] || [[C4|<math>C_4</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
| 4 || [[C2xC2|2]] || [[C2xC2|<math>C_2 \times C_2</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Er82], [Li94] ]] ||<br />
|- <br />
|8 || [[C8|1]] || [[C8|<math>C_8</math>]] ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|8 || [[C4xC2|2]] || [[C4xC2|<math>C_4 \times C_2</math>]] ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|8 || [[D8|3]] || [[D8|<math>D_8</math>]] ||6(?) || <math>k</math> || <math>k</math> || [[References|[Er87] ]] || Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] <br />
|-<br />
|8 || [[Q8|4]] || [[Q8|<math>Q_8</math>]] ||3(3) || <math>\mathcal{O}</math> || <math>k</math> || [[References|[Er88a], [Er88b], [HKL07], [Ei16]]] || <br />
|-<br />
|8 || [[C2xC2xC2|5]] || [[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]] || 8(8) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References| [Ea16]]] || Uses CFSG<br />
|}<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>|D|=16</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Donovan (w.r.t)?<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|16 || [[C16|1]] || [[C16|<math>C_{16}</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|16 || [[C4xC4|2]] || [[C4xC4|<math>C_4 \times C_4</math>]] || <math>\mathcal{O}</math> || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[EKKS14] ]] ||<br />
|-<br />
|16 || [[MNA(2,1)|3]] || [[MNA(2,1)]] || No || 3(?) || No || || [[References|[Sa11] ]] || Block invariants known<br />
|-<br />
|16 || [[C4:C4|4]] || [[C4:C4|<math>C_4:C_4</math>]] || <math>\mathcal{O}</math>|| 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[CG12], [Sa12b]]] ||<br />
|-<br />
|16 || [[C8xC2|5]] || [[C8xC2|<math>C_8 \times C_2</math>]]|| <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|16 || [[M16|6]] || [[M16|<math>M_{16}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[CG12], [Sa12b] ]] || <br />
|-<br />
|16 || [[D16|7]] || [[D16|<math>D_{16}</math>]] || <math>k</math>|| 5(?) || <math>k</math> || <math>k</math> || [[References|[Er87]]] || Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] <br />
|-<br />
|16 || [[SD16|8]] || [[SD16|<math>SD_{16}</math>]] || <math>k</math> || 7(?) || || || [[References|[Er88c], [Er90b]]] || Two other possible classes<br />
|-<br />
|16 || [[Q16|9]] || [[Q16|<math>Q_{16}</math>]] || No || 6(?) || || <math>k</math> || [[References|[Er88a], [Er88b], [Ho97]]] || Two possibly infinite families when <math>l(B)=2</math>. Classified over <math>\mathcal{O}</math> when <math>l(B)=3</math> in [[References#E|[Ei16]]]. Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20b]]]<br />
|-<br />
|16 || [[C4xC2xC2|10]] || [[C4xC2xC2|<math>C_4 \times C_2 \times C_2</math>]]|| <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[EL18a]]] ||<br />
|-<br />
|16 || [[D8xC2|11]] || [[D8xC2|<math>D_8 \times C_2</math>]] || No || || || || [[References|[Sa12] ]] || Block invariants known<br />
|-<br />
|16 || [[Q8xC2|12]] || [[Q8xC2|<math>Q_8 \times C_2</math>]] || <math>\mathcal{O}</math> || 3(3) || No || || [[References#E|[EL20]]] || Block invariants known by [[References#S|[Sa13]]]<br />
|-<br />
|16 || [[D8*C4|13]] || [[D8*C4|<math>D_8*C_4</math>]] || No || 3(?) || No || || [[References|[Sa13b] ]] || Block invariants known<br />
|-<br />
|16 || [[(C2)^4|14]] || [[(C2)^4|<math>(C_2)^4</math>]] || <math>\mathcal{O}</math> || 16(16) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Ea18] ]] ||<br />
|}<br />
<br />
The table for defect groups of order 32 takes as its starting point Table 13.1 of Sambale's book [[References|[Sa14]]].<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>|D|=32</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Donovan (w.r.t)?<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|32 || [[C32|1]] || [[C32|<math>C_{32}</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|32 || [[MNA(2,2)|2]] || [[MNA(2,2)|<math>MNA(2,2)</math>]] || <math>\mathcal{O}</math> || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EKS12]]] ||<br />
|-<br />
|32 || [[C8xC4|3]] || [[C8xC4|<math>C_8 \times C_4</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|-<br />
|32 || [[C8:C4|4]] || [[C8:C4|<math>C_8:C_4</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[MNA(3,1)|5]] || [[MNA(3,1)|<math>MNA(3,1)</math>]] || No || || || || [[References#S|[Sa11] ]] || Invariants known<br />
|-<br />
|32 || [[MNA(2,1):C2|6]] || [[MNA(3,1):C2|<math>MNA(2,1):C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,7)|7]] || [[SmallGroup(32,7)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] || <math>M_{16}:C_2</math><br />
|-<br />
|32 || [[2.MNA(2,1)|8]] || [[2.MNA(2,1)|<math>2.MNA(2,1)</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[D8:C4|9]] || [[D8:C4|<math>D_8:C_4</math>]] || No || || || || [[References#S|[Sa14,10.23]]] || Invariants known<br />
|-<br />
|32 || [[Q8:C4|10]] || [[Q8:C4|<math>Q_8:C_4</math>]] || No || || || || [[References#S|[Sa14,10.25]]] || Invariants known<br />
|-<br />
|32 || [[C4wrC2|11]] || [[C4wrC2|<math>C_4 \wr C_2</math>]] || No || 6(6) || No || || [[References#K|[Ku80]]], [[References#K|[KoLaSa23]]] || Invariants known. Principal blocks classified up to source algebra equivalence in [[References#K|[KoLaSa23]]]<br />
|-<br />
|32 || [[C4:C8|12]] || [[C4:C8|<math>C_4:C_8</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[C8:C4a|13]] || [[C8:C4a|<math>C_8:C_4=\langle a,b|a^8=b^4=1, ba=a^3b \rangle</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[C8:C4b|14]] || [[C8:C4b|<math>C_8:C_4=\langle a,b|a^8=b^4=1, ba=a^7b \rangle</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,15)|15]] || [[SmallGroup(32,15)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[C16xC2|16]] || [[C16xC2|<math>C_{16} \times C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|32 || [[M32|17]] || [[M32|<math>M_{32}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[D32|18]] || [[D32|<math>D_{32}</math>]] || <math>k</math> || 5(?) || <math>k</math> || <math>k</math> || [[References#E|[Er87]]] || Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] <br />
|-<br />
|32 || [[SD32|19]] || [[SD32|<math>SD_{32}</math>]] || <math>k</math> || || || || ||<br />
|-<br />
|32 || [[Q32|20]] || [[Q32|<math>Q_{32}</math>]] || No || || || || [[References#E|[Er88a], [Er88b], [Ho97]]] || Two possibly infinite families when <math>l(B)=2</math>. Classified over <math>\mathcal{O}</math> when <math>l(B)=3</math> in [[References#E|[Ei16]]]. Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20b]]]<br />
|-<br />
|32 || [[C4xC4xC2|21]] || [[C4xC4xC2|<math>C_4 \times C_4 \times C_2</math>]] || <math>\mathcal{O}</math> || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EKKS14]]] <br />
|-<br />
|32 || [[MNA(2,1)xC2|22]] || [[MNA(2,1)xC2|<math>MNA(2,1) \times C_2</math>]] || No || || || || [[References#S|[Sa14,10.25]]] || Invariants known<br />
|-<br />
|32 || [[(C4:C4)xC2|23]] || [[(C4:C4)xC2|<math>(C_4:C_4) \times C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,24)|24]] || [[SmallGroup(32,24)]]<!--<math>(C_4 \times C_4):C_2=\langle a,b,c \mid a^4 = b^4 = c^2 = e, ab = ba, ac = ca, cb = a^2bc \rangle</math>]]--> || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[D8xC4|25]] || [[D8xC4|<math>D_8 \times C_4</math>]] || No || || || || [[References#S|[Sa14,9.7]]] ||<br />
Invariants known<br />
|-<br />
|32 || [[Q8xC4|26]] || [[Q8xC4|<math>Q_8 \times C_4</math>]] || <math>\mathcal{O}</math> || 3(3) || No || || [[References#E|[EL20]]] || Invariants known by [[References#S|[Sa14,9.28]]]<br />
|-<br />
|32 || [[SmallGroup(32,27)|27]] || [[SmallGroup(32,27)]]<!--|<math>(C_4 \times C_4):C_2=\langle x,y,z,a,b \mid x^2 = y^2 = z^2 = a^2 = b^2 = e, xy = yx, xz, = zx, yz = zy, aza^{-1} = xz, bzb^{-1} = yz, ax = xa, ay = ya, bx = xb, by = yb \rangle</math>]]--> || No || || || || ||<br />
|-<br />
|32 || [[SmallGroup(32,28)|28]] || [[SmallGroup(32,28)]] || No || || || || [[References#S|[Sa14,13.11]]] || Invariants known<br />
|-<br />
|32 || [[SmallGroup(32,29)|29]] || [[SmallGroup(32,29)]] || No || || || || [[References#S|[Sa14,13.11]]] || Invariants known<br />
|-<br />
|32 || [[SmallGroup(32,30)|30]] || [[SmallGroup(32,30)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,31)|31]] || [[SmallGroup(32,31)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,32)|32]] || [[SmallGroup(32,32)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,33)|33]] || [[SmallGroup(32,33)]] || No || || || || [[References|[Sa14,13.12]]] || Invariants partly known<br />
|-<br />
|32 || [[SmallGroup(32,34)|34]] || [[SmallGroup(32,34)]] || No || || || || [[References|[Sa14,13.12]]] || Invariants partly known<br />
|-<br />
|32 || [[C4:Q8|35]] || [[C4:Q8|<math>C_4:Q_8</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[C8xC2xC2|36]] || [[C8xC2xC2|<math>C_8 \times C_2 \times C_2</math>]] || <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EL18a]]] ||<br />
|-<br />
|32 || [[M16xC2|37]] || [[M16xC2|<math>M_{16} \times C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[D8*C8|38]] || [[D8*C8|<math>D_8*C_8</math>]] || No || || || || [[References#S|[Sa14,9.18]]] || Invariants known<br />
|-<br />
|32 || [[D16xC2|39]] || [[D16xC2|<math>D_{16} \times C_2</math>]] || No || || || || [[References#S|[Sa14,9.7]]] || Invariants known<br />
|-<br />
|32 || [[SD16xC2|40]] || [[SD16xC2|<math>SD_{16} \times C_2</math>]] || No || || || || [[References#S|[Sa14,9.37]]] || Invariants known<br />
|-<br />
|32 || [[Q16xC2|41]] || [[Q16xC2|<math>Q_{16} \times C_2</math>]] || No || || || || [[References#S|[Sa14,9.28]]] || Invariants known<br />
|-<br />
|32 || [[D16*C4|42]] || [[D16*C4|<math>D_{16}*C_4</math>]] || No || || || || [[References|[Sa14,9.18]]] || Invariants known<br />
|-<br />
|32 || [[SmallGroup(32,43)|43]] || [[SmallGroup(32,43)]] || No || || || || ||<br />
|-<br />
|32 || [[SmallGroup(32,44)|44]] || [[SmallGroup(32,44)]] || No || || || || ||<br />
|-<br />
|32 || [[C4xC2xC2xC2|45]] || [[C4xC2xC2xC2|<math>C_4 \times C_2 \times C_2 \times C_2</math>]] || <math>\mathcal{O}</math> || 8(8) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EL23]]] ||<br />
|-<br />
|32 || [[D8xC2xC2|46]] || [[D8xC2xC2|<math>D_8 \times C_2 \times C_2</math>]] || No || || || || ||<br />
|-<br />
|32 || [[Q8xC2xC2|47]] || [[Q8xC2xC2|<math>Q_8 \times C_2 \times C_2</math>]] || No || || || || ||<br />
|-<br />
|32 || [[D8*C4xC2|48]] || [[D8*C4xC2|<math>(D_8*C_4) \times C_2</math>]] || No || || || || ||<br />
|-<br />
|32 || [[D8*D8|49]] || [[D8*D8|<math>D_8*D_8</math>]] || No || || || || [[References#S|[Sa13c]]] || Invariants partly known<br />
|-<br />
|32 || [[D8*Q8|50]] || [[D8*Q8|<math>D_8*Q_8</math>]] || No || || || || [[References#S|[Sa13c]]] || Invariants partly known<br />
|-<br />
|32 || [[(C2)^5|51]] || [[(C2)^5|<math>(C_2)^5</math>]] || <math>\mathcal{O}</math> || 34 (34) || <math>\mathcal{O}</math> || || [[References#A|[Ar19]]] || Derived eq. classes determined for 30 of the 34 Morita eq. classes. <br />
|}<br />
<br />
<!--<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>|D|=64</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Donovan (w.r.t)?<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|64 || [[C64|1]] || [[C64|<math>C_{64}</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
|64 || [[C8xC8|2]] || [[C8xC8|<math>C_8 \times C_8</math>]] || <math>\mathcal{O}</math> ||2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EKKS14]]]||<br />
|-<br />
|64 || [[SmallGroup(64,3)|3]] || [[SmallGroup(64,3)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_8:C_8</math><br />
|-<br />
|64 || [[(C2xC2xC2):C8|4]] || [[(C2xC2xC2):C8|<math>(C_2)^3:C_8</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,5)|5]] || [[SmallGroup(64,5)]] || No || || || || ||<br />
|-<br />
|64 || [[(D8:C8|6]] || [[D8:C8|<math>D_8:C_8</math>]] || No || || || || ||<br />
|-<br />
|64 || [[(Q8:C8|7]] || [[Q8:C8|<math>Q_8:C_8</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,8)|8]] || [[SmallGroup(64,8)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,9)|9]] || [[SmallGroup(64,9)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,10)|10]] || [[SmallGroup(64,10)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,11)|11]] || [[SmallGroup(64,11)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,12)|12]] || [[SmallGroup(64,12)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,13)|13]] || [[SmallGroup(64,13)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,14)|14]] || [[SmallGroup(64,14)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,15)|15]] || [[SmallGroup(64,15)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_8:C_8</math><br />
|-<br />
|64 || [[SmallGroup(64,16)|16]] || [[SmallGroup(64,16)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_8:C_8</math><br />
|-<br />
|64 || [[SmallGroup(64,17)|17]] || [[SmallGroup(64,17)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,18)|18]] || [[SmallGroup(64,18)]] || No || || || || || <math>(C_4 \times C_4):C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,19)|19]] || [[SmallGroup(64,19)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,20)|20]] || [[SmallGroup(64,20)]] || No || || || || || <math>(C_4 \times C_4):C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,21)|21]] || [[SmallGroup(64,21)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,22)|22]] || [[SmallGroup(64,22)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,23)|23]] || [[SmallGroup(64,23)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,24)|24]] || [[SmallGroup(64,24)]] || No || || || || || <math>M_{16}:C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,25)|25]] || [[SmallGroup(64,25)]] || No || || || || || <math>M_{16}:C_4</math><br />
|-<br />
|64 || [[C16xC4|26]] || [[C16xC4|<math>C_{16} \times C_4</math>]]|| <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|64 || [[SmallGroup(64,27)|27]] || [[SmallGroup(64,27)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_{16}:C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,28)|28]] || [[SmallGroup(64,28)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_{16}:C_4</math><br />
|-<br />
|64 || [[(C2xC2):C16|29]] || [[(C2xC2):C16|<math>(C_2)^2:C_{16}</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,30)|30]] || [[SmallGroup(64,30)]] || No || || || || || <math>M_{32}:C_2</math><br />
|-<br />
|64 || [[SmallGroup(64,31)|31]] || [[SmallGroup(64,31)]] || No || || || || || <math>M_{32}:C_2</math><br />
|-<br />
|64 || [[C2wrC4|32]] || [[(C2wrC4|<math>C_2 \wr C_4</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,33)|33]] || [[SmallGroup(64,33)]] || No || || || || || <br />
|-<br />
|64 || [[SmallGroup(64,34)|34]] || [[SmallGroup(64,31)]] || No || || || || || <math>(C_4 \times C_4):C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,35)|35]] || [[SmallGroup(64,35)]] || No || || || || || <math>(C_4 \times C_4):C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,36)|36]] || [[SmallGroup(64,36)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,37)|37]] || [[SmallGroup(64,37)]] || No || || || || || <math>C_4:Q_8</math>, fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,38)|38]] || [[SmallGroup(64,38)]] || No || || || || || <math>D_{16}:C_4</math>, fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,39)|39]] || [[SmallGroup(64,39)]] || No || || || || || <math>Q_{16}:C_2</math><br />
|-<br />
|64 || [[SmallGroup(64,40)|40]] || [[SmallGroup(64,40)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,79)|79]] || [[SmallGroup(64,79)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|-<br />
|64 || [[SmallGroup(64,81)|81]] || [[SmallGroup(64,81)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|}<br />
--><br />
<br />
==Blocks for <math>p=3</math>==<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>1 \leq |D| \leq 27</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|- <br />
| 3 || [[C3|1]] || [[C3|<math>C_3</math>]] || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
|9 || [[C9|1]] ||[[C9|<math>C_9</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|-<br />
|9 || [[C3xC3|2]] || [[C3xC3|<math>C_3 \times C_3</math>]] || || || || ||<br />
|-<br />
|27 || [[C27|1]] || [[C27|<math>C_{27}</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|27 || [[C9xC3|2]] || [[C9xC3|<math>C_9 \times C_3</math>]] || || || || ||<br />
|-<br />
|27 || [[3_+^3|3]] || [[3_+^3|<math>3_+^{1+2}</math>]] || || || || ||<br />
|-<br />
|27 || [[3_-^3|4]] || [[3_-^3|<math>3_-^{1+2}</math>]] || || || || ||<br />
|-<br />
|27 || [[C3xC3xC3|5]] || [[C3xC3xC3|<math>C_3 \times C_3 \times C_3</math>]] || || || || ||<br />
|}<br />
<br />
==Blocks for <math>p=5</math>==<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>5 \leq |D| \leq 125</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|- <br />
|5 || [[C5|1]] || [[C5|<math>C_5</math>]] ||6(6) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
|25 || [[C25|1]] ||[[C25|<math>C_{25}</math>]] || 6(6) || No || <math>\mathcal{O}</math> || || Max 12 classes <br />
|-<br />
|25 || [[C5xC5|2]] || [[C5xC5|<math>C_5 \times C_5</math>]] || || || || ||<br />
|- <br />
|125 || [[C125|1]] ||[[C125|<math>C_{125}</math>]] || || || || || <br />
|-<br />
|125 || [[C25xC5|2]] || [[C25xC5|<math>C_{25} \times C_5</math>]] || || || || || <br />
|-<br />
|125 || [[5_+^3|3]] || [[5_+^3|<math>5_+^{1+2}</math>]] || 62(62) || <math>\mathcal{O}</math> || || [[References#A|[AE23]]] || Inertial quotients are consistent within classes<br />
|-<br />
|125 || [[5_-^3|4]] || [[5_-^3|<math>5_-^{1+2}</math>]] || || || || ||<br />
|-<br />
|125 || [[C5xC5xC5|5]] || [[C5xC5xC5|<math>C_5 \times C_5 \times C_5</math>]] || || || || ||<br />
|}<br />
<br />
==Blocks for <math>p\geq 7</math>==<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>|D|</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|- <br />
|7 || [[C7|1]] || [[C7|<math>C_7</math>]] ||14(14) ||No || <math>\mathcal{O}</math> || ||Max 21 classes <br />
|- <br />
|11|| [[C11|1]] || [[C11|<math>C_{11}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|- <br />
|13 || [[C13|1]] || [[C13|<math>C_{13}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|-<br />
|17|| [[C17|1]] || [[C17|<math>C_{17}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|- <br />
|19 || [[C19|1]] || [[C19|<math>C_{19}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|-<br />
|23 || [[C23|1]] || [[C23|<math>C_{23}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|}</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Generic_classifications_by_p-group_class&diff=1231
Generic classifications by p-group class
2023-10-23T17:00:50Z
<p>Charles Eaton: <math>C_{2^n} \wr C_2</math></p>
<hr />
<div>This page contains results for classes of ''p''-groups for which we either have classifications or have general results concerning Morita equivalence classes. <br />
<br />
== Fusion trivial ''p''-groups ==<br />
<br />
''p''-groups <math>P</math> for which the only saturated fusion system is <math>\mathcal{F}_P(P)</math> have not yet been given a name in the literature (to our knowledge). We will call them ''fusion trivial'', but ''nilpotent forcing'' also seems appropriate following [[References#V|[vdW91]]] (where ''p''-groups for which any finite group <math>G</math> containing <math>P</math> as a Sylow ''p''-subgroup must be ''p''-nilpotent are called ''p-nilpotent forcing''). It is not known whether these two definitions are equivalent, i.e., whether there exist ''p''-nilpotent forcing ''p''-groups for which there is an exotic fusion system.<br />
<br />
Blocks with fusion trivial defect groups must be nilpotent and so Morita equivalent to the group algebra of a defect group by [[References#P|[Pu88]]].<br />
<br />
Examples of fusion trivial ''p''-groups are abelian ''2''-groups with automorphism group a ''2''-group (i.e., those whose cyclic factors have pairwise distinct orders), and metacyclic 2-groups other than homocyclic, dihedral, generalised quaternion or semidihedral groups (see [[References#C|[CG12]]] or [[References#S|[Sa12b]]]).<br />
<br />
Note that a ''p''-group is fusion trivial if and only if it is resistant and has automorphism group a ''p''-group. See [[References#S|[St06]]] for an analysis of resistant ''p''-groups.<br />
<br />
== Cyclic ''p''-groups ==<br />
<br />
[[Blocks with cyclic defect groups|Click here for background on blocks with cyclic defect groups]].<br />
<br />
Morita equivalence classes are labelled by [[Brauer trees]], but it is at present an open problem as to which Brauer trees are realised by blocks of finite groups. Each ''k''-Morita equivalence class corresponds to an unique <math>\mathcal{O}</math>-Morita equivalence class. <br />
<br />
For <math>p=2,3</math> every appropriate Brauer tree is realised by a block and we can give generic descriptions.<br />
<br />
[[C(2^n)|<math>2</math>-blocks with cyclic defect groups]]<br />
<br />
[[C(3^n)|<math>3</math>-blocks with cyclic defect groups]]<br />
<br />
== Tame blocks ==<br />
<br />
[[Tame blocks|Click here for background on tame blocks]].<br />
<br />
Erdmann classified algebras which are candidates for [[Basic algebras|basic algebras]] of tame blocks, i.e., those with dihedral, semidihedral or generalised quaternion defect groups (see [[References|[Er90] ]]) and in the cases of dihedral and semihedral defect groups determined which are realised by blocks of finite groups. In the case of generalised quaternion groups, the case of blocks with two simple modules is still open. These classifications only hold with respect to the field ''k'' at present.<br />
<br />
Principal blocks with dihedral defect groups are classified up to source algebra equivalence in [[References#K|[KoLa20]]].<br />
<br />
== Abelian <math>2</math>-groups with <math>2</math>-rank at most four ==<br />
<br />
[[Image:under-construction.png|50px|left]]<br />
<br />
These have been classified in [[References#W|[WZZ18]]], [[References#E|[EL18a]]] and [[References#E|[EL23]]] with respect to <math>\mathcal{O}</math>. The derived equivalences classes with respect to <math>\mathcal{O}</math> are known.<br />
<br />
Let <math>l,m,n,r \geq 2</math> be distinct.<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>rk_2(D) \leq 4</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>D</math> <br />
! scope="col"| # <math>(\mathcal{O}</math>-)Morita classes<br />
! scope="col"| # <math>\mathcal{O}</math>-Derived equiv classes<br />
! scope="col"| Always endopermutation source Morita equivalent?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|[[C(2^n)|<math>C_{2^n}</math>]] || 1 || 1 || Yes || [[References#L|[Li96]]] || [[Blocks with cyclic defect groups]]<br />
|-<br />
|[[C2xC2|<math>C_2 \times C_2</math>]] || 3 || 2 || Yes || [[References#L|[Li94]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)|<math>C_{2^m} \times C_{2^m}</math>]] || 2 || 2 || Yes || [[References#E|[EKKS14]]] ||<br />
|-<br />
|[[C(2^m)xC(2^n)|<math>C_{2^m} \times C_{2^n}</math>]] || 1 || 1 || Yes || [[References#P|[Pu88]]] || [[Nilpotent blocks]]<br />
|-<br />
|[[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]] || 8 || 4 || || [[References#E|[Ea16]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)xC(2^m)|<math>C_{2^m} \times C_{2^m} \times C_{2^m}</math>]] || 4 || 4 || Yes || [[References#W|[WZZ18]]] ||<br />
|-<br />
|[[C(2^m)xC2xC2|<math>C_{2^m} \times C_2 \times C_2</math>]] || 3 || 2 || || [[References#E|[EL18a]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)xC(2^n)|<math>C_{2^m} \times C_{2^m} \times C_{2^n}</math>]] || 2 || 2 || || [[References#W|[WZZ18]]] ||<br />
|-<br />
|[[C(2^l)xC(2^m)xC(2^n)|<math>C_{2^l} \times C_{2^m} \times C_{2^n}</math>]] || 1 || 1 || Yes || [[References#P|[Pu88]]] || [[Nilpotent blocks]]<br />
|-<br />
|[[(C2)^4|<math>(C_2)^4</math>]] || 16 || || || [[References#E|[Ea18]]] ||<br />
|-<br />
|[[C(2^m)xC2xC2xC2|<math>C_{2^m} \times C_2 \times C_2 \times C_2</math>]] || || || || [[References#E|[EL23]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)xC2xC2|<math>C_{2^m} \times C_{2^m} \times C_2 \times C_2</math>]] || || || || [[References#E|[EL23]]] ||<br />
|-<br />
|[[C(2^m)xC(2^n)xC2xC2|<math>C_{2^m} \times C_{2^n} \times C_2 \times C_2</math>]] || || || || [[References#E|[EL23]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)xC(2^n)xC(2^n)|<math>C_{2^m} \times C_{2^m} \times C_{2^n} \times C_{2^n}</math>]] || || || || [[References#E|[EL23]]] ||<br />
|-<br />
|[[C(2^l)xC(2^l)xC(2^m)xC(2^n)|<math>C_{2^l} \times C_{2^l} \times C_{2^m} \times C_{2^n}</math>]] || || || || [[References#E|[EL23]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)xC(2^m)xC(2^n)|<math>C_{2^m} \times C_{2^m} \times C_{2^m} \times C_{2^n}</math>]] || || || || [[References#E|[EL23]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)xC(2^m)xC(2^m)|<math>C_{2^m} \times C_{2^m} \times C_{2^m} \times C_{2^m}</math>]] || || || || [[References#E|[EL23]]] ||<br />
|-<br />
|[[C(2^l)xC(2^m)xC(2^n)xC(2^r)|<math>C_{2^l} \times C_{2^m} \times C_{2^n} \times C_{2^n}</math>]] || || || || [[References#E|[EL23]]] || [[Nilpotent blocks]] <br />
|}<br />
<br />
<br />
<!--== Abelian <math>2</math>-groups ==<br />
<br />
Donovan's conjecture holds for ''2''-blocks with abelian defect groups. Some generic classification results are known for certain inertial quotients. These will be detailed here.--><br />
<br />
== Minimal nonabelian <math>2</math>-groups ==<br />
<br />
Blocks with defect groups which are minimal nonabelian <math>2</math>-groups of the form <math>P=\langle x,y:x^{2^r}=y^{2^r}=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle</math> are classified in [[References#E|[EKS12]]]. There are two <math>\mathcal{O}</math>-Morita equivalence classes, with representatives <math>\mathcal{O}P</math> and <math>\mathcal{O}(P:C_3)</math>.<br />
<br />
For arbitrary minimal nonabelian <math>2</math>-groups, by [[References#S|[Sa16]]] blocks with such defect groups and the same fusion system are isotypic.<br />
<br />
== Homocyclic <math>2</math>-groups when inertial quotient contains a Singer cycle == <br />
<br />
A Singer cycle is an element of order <math>p^n-1</math> in <math>\operatorname{Aut}({C_p}^n) \cong GL_n(p)</math>, and a subgroup generated by a Singer cycle acts transitively on the non-trivial elements of <math>(C_p)^n</math>.<br />
<br />
<math>2</math>-blocks with homocyclic defect group <math> D \cong (C_{2^m})^n </math> whose inertial quotient <math> \mathbb{E} </math> contains a Singer cycle are classified in [[References#E|[McK19]]]. In this situation, <math>\mathbb{E}</math> has the form <math>E:F </math> where <math>E \cong C_{2^n-1}</math> and <math>F</math> is trivial or a subgroup of <math>C_n</math>. There are three <math> \mathcal{O} </math>-Morita equivalence classes when <math>m=1, n =3 </math>; two when <math>m=1 , n \neq 3 </math>; and only one when <math>m> 1 </math>. The three classes have representatives [[M(8,5,7) | <math> \mathcal{O}J_1 </math>]], which occurs only when <math>m=1, n=3</math>; <math> \mathcal{O}(SL_2(2^n):F) </math>, which occurs only when <math>m=1 </math>; and <math> \mathcal{O}(D : \mathbb{E})</math>. <br />
<br />
The Morita equivalence between the block and the class representative is known to be basic, possibly except when <math>m=1,n=3</math>, since the Morita equivalences between the [[M(8,5,8) |principal block of <math>{\rm \operatorname{Aut}(SL_2(8))}</math>]] and the blocks [[M(8,5,8)#Other_notatable_representatives |<math>B_0(\mathcal{O}({}^2G_2(3^{2m+1})))</math>]] are not known to be basic.<br />
<br />
<br />
== Abelian <math>2</math>-groups when inertial quotient is cyclic and acts freely on the defect group == <br />
<br />
<math>2</math>-blocks with abelian defect group <math> D </math> whose inertial quotient <math> E </math> is a cyclic group that acts freely on the defect group (i.e. such that the stabiliser in <math> E </math> of any nontrivial element of <math>D</math> is trivial) are classified in [[References#E|[ArMcK20]]].<br />
<br />
If the action of <math>E</math> is transitive, then <math>D</math> is homocyclic, <math>E</math> is a Singer cycle and, by the previous section, the block is either Morita equivalent to the principal block of <math> \mathcal{O}SL_2(2^n) </math>, or to <math> \mathcal{O}(D : E)</math>. <br />
<br />
If the action of <math>E</math> is not transitive, then the block is Morita equivalent to <math> \mathcal{O}(D : E)</math>. <br />
<br />
In each case, the Morita equivalence between the block and the class representative is known to be basic.<br />
<br />
== Extraspecial <math>p</math>-groups of order <math>p^3</math> and exponent <math>p</math> for <math>p \geq 5</math> ==<br />
<br />
These blocks are described in [[References#A|[AE23]]]. Donovan's conjecture holds in this case.<br />
<br />
== Principal blocks with wreath products <math>C_{2^n} \wr C_2</math> defect groups ==<br />
<br />
These are classified up to splendid Morita equivalence in [[References#K|[KoLaSa23]]]. There are six classes for each <math>n</math>, with representatives the principal blocks of:<br />
<br />
<math>C_{2^n} \wr C_2</math><br />
<br />
<math>(C_{2^n} \times C_{2^n}):S_3</math><br />
<br />
<math>SL_2^n(q)</math> where <math>(q-1)_2=2^n</math>, consisting of matrices whose determinant is a <math>2^n</math> root of unity<br />
<br />
<math>SU_2^n(q)</math> where <math>(q+1)_2=2^n</math>, consisting of matrices whose determinant is a <math>2^n</math> root of unity <br />
<br />
<math>PSL_3(q)</math> where <math>(q-1)_2=2^n</math><br />
<br />
<math>PSU_3(q)</math> where <math>(q+1)_2=2^n</math></div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=References&diff=1230
References
2023-10-23T16:49:08Z
<p>Charles Eaton: [KoLaSa23]</p>
<hr />
<div>[[#A|A,]] [[#B|B,]] [[#C|C,]] [[#D|D,]] [[#E|E,]] [[#F|F,]] [[#G|G,]] [[#H|H,]] [[#I|I,]] [[#J|J,]] [[#K|K,]] [[#L|L,]] [[#M|M,]] [[#N|N,]] [[#O|O,]] [[#P|P,]] [[#Q|Q,]] [[#R|R,]] [[#S|S,]] [[#T|T]] [[#U|U,]] [[#V|V,]] [[#W|W,]] [[#X|X,]] [[#Y|Y,]] [[#Z|Z,]] <br />
<br />
{| <br />
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|-<br />
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|-<br />
|[AE04] || '''Jianbei An and C. W. Eaton''', ''Blocks with trivial intersection defect groups'', Math. Z. '''247''' (2004), 461-486.<br />
|-<br />
|[AE04] || '''Jianbei An and C. W. Eaton''', [https://arxiv.org/abs/2310.02150 ''Morita equivalence classes of blocks with extraspecial defect groups <math>p_+^{1+2}</math>''], [https://arxiv.org/abs/2310.02150 arxiv:2310.02150]<br />
|-<br />
|[Ar19] || '''C. G. Ardito''', [https://arxiv.org/abs/1908.02652 ''Morita equivalence classes of blocks with elementary abelian defect groups of order 32''], J. Algebra '''573''' (2021), 297-335.<br />
|-<br />
|[ArMcK20] || '''C. G. Ardito and E. McKernon''', ''[https://arxiv.org/abs/2010.08329 ''2-blocks with an abelian defect group and a freely acting cyclic inertial quotient''], [https://arxiv.org/abs/2010.08329 arxiv.org/abs/2010.08329]<br />
|-<br />
|[AS20] || '''C. G. Ardito and B. Sambale''', [http://www.advgrouptheory.com/journal/Volumes/12/ArditoSambale.pdf ''Broué's Conjecture for 2-blocks with elementary abelian defect groups of order 32''], Advances in Group Theory and Applications 12 (2021), 71–78. <br />
|-<br />
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|-<br />
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|-<br />
|[BKL18] || '''R. Boltje, R. Kessar, and M. Linckelmann''', [https://doi.org/10.1016/j.jalgebra.2019.02.045 ''On Picard groups of blocks of finite groups''], J. Algebra '''558''' (2020), 70-101.<br />
|-<br />
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|-<br />
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|-<br />
|[CDR18] || '''D. A. Craven, O. Dudas and R. Rouquier''', [https://arxiv.org/abs/1701.07097 ''The Brauer trees of unipotent blocks''], to appear, J. EMS, [https://arxiv.org/abs/1701.07097 arXiv:1701.07097] <br />
|-<br />
|[CEKL11] || '''D. A. Craven, C. W. Eaton, R. Kessar and M. Linckelmann''', ''The structure of blocks with a Klein four defect group'', Math. Z. '''268''' (2011), 441-476.<br />
|-<br />
|[CG12] || '''D. A. Craven and A. Glesser''', ''Fusion systems on small p-groups'', Trans. AMS '''364''' (2012) 5945-5967.<br />
|-<br />
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|-<br />
|[CuRe81a] || '''C. W. Curtis and I. Reiner''', ''Methods of representation theory, with applications to finite groups and orders, Volume I'', Wiley-Interscience (1981).<br />
|-<br />
|[CuRe81b] || '''C. W. Curtis and I. Reiner''', ''Methods of representation theory, with applications to finite groups and orders, Volume II'', Wiley-Interscience (1981).<br />
|- id="D"<br />
|[Da66] || '''E. C. Dade''', ''Blocks with cyclic defect groups'', Ann. Math. '''84''' (1966), 20-48. <br />
|-<br />
|[DE20] || '''S. Danz and K. Erdmann''', [https://arxiv.org/abs/2008.10999 ''On Ext-Quivers of Blocks of weight two for symmetric groups''], [https://arxiv.org/abs/2008.10999 arXiv:2008.10999]<br />
|-<br />
|[Du14] || '''O. Dudas''', [https://arxiv.org/abs/1011.5478 ''Coxeter orbits and Brauer trees II''], Int. Math. Res. Not. '''15''' (2014), 4100-4123.<br />
|-<br />
|[Dü04] || '''O. Düvel''', ''On Donovan's conjecture'', J. Algebra '''272''' (2004), 1-26.<br />
|- id="E"<br />
|[Ea16] || '''C. W. Eaton''', ''Morita equivalence classes of <math>2</math>-blocks of defect three'', Proc. AMS '''144''' (2016), 1961-1970.<br />
|-<br />
|[Ea18] || '''C. W. Eaton''', [https://arxiv.org/abs/1612.03485 ''Morita equivalence classes of blocks with elementary abelian defect groups of order 16''], [https://arxiv.org/abs/1612.03485 arXiv:1612.03485]<br />
|-<br />
|[EEL18] || '''C. W. Eaton, F. Eisele and M. Livesey''', [https://arxiv.org/abs/1809.08152 ''Donovan's conjecture, blocks with abelian defect groups and discrete valuation rings''], Math. Z. '''295''' (2020), 249-264.<br />
|-<br />
|[EKKS14] || '''C. W. Eaton, R. Kessar, B. Külshammer and B. Sambale''', ''<math>2</math>-blocks with abelian defect groups'', Adv. Math. '''254''' (2014), 706-735.<br />
|-<br />
|[EKS12] || '''C. W. Eaton, B. Külshammer and B. Sambale''', ''<math>2</math>-blocks with minimal nonabelian defect groups, II'', J. Group Theory '''15''' (2012), 311-321.<br />
|-<br />
|[EL18a] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/1709.04331 Classifying blocks with abelian defect groups of rank 3 for the prime 2]'', J. Algebra '''515''' (2018), 1-18.<br />
|-<br />
|[EL18b] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/1803.03539 Donovan's conjecture and blocks with abelian defect groups]'', Proc. AMS. '''147''' (2019), 963-970.<br />
|-<br />
|[EL18c] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/1810.10950 Some examples of Picard groups of blocks]'', J. Algebra '''558''' (2020), 350-370.<br />
|-<br />
|[EL20] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/2006.11173 Donovan's conjecture and extensions by the centralizer of a defect group]'', J. Algebra '''582''' (2021), 157-176.<br />
|-<br />
|[EL23] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/2310.05734 Morita equivalence classes of <math>2</math>-blocks with abelian defect groups of rank <math>4</math>]'', [https://arxiv.org/abs/2310.05734 arxiv:2310.05734]<br />
|-<br />
|[Ei16] || '''F. Eisele''', ''Blocks with a generalized quaternion defect group and three simple modules over a <math>2</math>-adic ring'', J. Algebra '''456''' (2016), 294-322.<br />
|-<br />
|[Ei18] || '''F. Eisele''', ''[https://arxiv.org/abs/1807.05110 The Picard group of an order and Külshammer reduction]'', Algebr. Represent. Theory '''24''' (2021), 505-518.<br />
|-<br />
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|-<br />
|[EiLiv20] || '''F. Eisele and M. Livesey''', ''[https://arxiv.org/abs/2006.13837 Arbitrarily large Morita Frobenius numbers]'', Algebra Number Theory '''16''' (2022), 1889-1904.<br />
|-<br />
|[Er82] || '''K. Erdmann''', ''Blocks whose defect groups are Klein four groups: a correction'', J. Algebra '''76''' (1982), 505-518.<br />
|-<br />
|[Er87] || '''K. Erdmann''', ''Algebras and dihedral defect groups'', Proc. LMS '''54''' (1987), 88-114.<br />
|-<br />
|[Er88a] || '''K. Erdmann''', ''Algebras and quaternion defect groups, I'', Math. Ann. '''281''' (1988), 545-560.<br />
|-<br />
|[Er88b] || '''K. Erdmann''', ''Algebras and quaternion defect groups, II'', Math. Ann. '''281''' (1988), 561-582. <br />
|-<br />
|[Er88c] || '''K. Erdmann''', ''Algebras and semidihedral defect groups I'', Proc. LMS '''57''' (1988), 109-150. <br />
|-<br />
|[Er90] || ''' K. Erdmann''', ''Blocks of tame representation type and related algebras'', Lecture Notes in Mathematics '''1428''', Springer-Verlag (1990).<br />
|-<br />
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|- id="F"<br />
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|-<br />
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|- id="G"<br />
|[GMdelR21] || '''D. Garcia, l. Margolis and A. del Rio''', [https://arxiv.org/abs/2016.07231 ''Non-isomorphic 2-groups with isomorphic modular group algebras''], J. Reine Angew. Math. '''f783''' (2022), 269–274.<br />
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|[GO97] || '''H. Gollan and T. Okuyama''', ''Derived equivalences for the smallest Janko group'', preprint (1997).<br />
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|-<br />
|[KoLaSa23] || '''S. Koshitani, C. Lassueur and B. Sambale''', [https://arxiv.org/abs/2310.13621 ''Principal <math>2</math>-blocks with wreathed defect groups up to splendid Morita equivalence''], [https://arxiv.org/abs/2310.13621 arxiv:2310.13621]<br />
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|-<br />
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|-<br />
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|-<br />
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|-<br />
|[LiMa20] || '''M. Livesey and C. Marchi''', [https://arxiv.org/abs/2002.10571 ''On Picent for blocks with normal defect group''], [https://arxiv.org/abs/2002.10571 arXiv:2002.10571]<br />
|-<br />
|[LiMa20b] || '''M. Livesey and C. Marchi''', [https://arxiv.org/abs/2008.05857 ''Picard groups for blocks with normal defect groups and linear source bimodules''], [https://arxiv.org/abs/2008.05857 arXiv:2008.05857]<br />
|- id="M"<br />
|[Mac] || '''N. Macgregor''', ''Morita equivalence classes of tame blocks of finite groups'', J. Algebra '''608''' (2022), 719-754.<br />
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|-<br />
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|[NS18] || '''G. Navarro and B. Sambale''', ''On the blockwise modular isomorphism problem'', Manuscripta Math. '''157''' (2018), 263-278.<br />
|- <br />
|[Ne02] || '''G. Nebe''', [http://www.math.rwth-aachen.de/~Gabriele.Nebe/papers/survey.pdf ''Group rings of finite groups over p-adic integers, some examples''], Proceedings of the conference Around Group rings (Edmonton) Resenhas '''5''' (2002), 329-350.<br />
|- id="O"<br />
|[Ok97] || '''T. Okuyama''', ''Some examples of derived equivalent blocks of finite groups'', preprint (1997).<br />
|- id="P"<br />
|[Pu88]|| '''L. Puig''', ''Nilpotent blocks and their source algebras'', Invent. Math. '''93''' (1988), 77-116.<br />
|-<br />
|[Pu94] || '''L. Puig''', ''On Joanna Scopes’ criterion of equivalence for blocks of symmetric groups'', Algebra Colloq. '''1''' (1994), 25-55.<br />
|-<br />
|[Pu99]|| '''L. Puig''', ''On the local structure of Morita and Rickard equivalences between Brauer blocks'', Progress in Math. '''178''', Birkhauser Verlag (1999).<br />
|-<br />
|[Pu09] || '''L. Puig''', ''Block source algebras in p-solvable groups'', Michigan Math. J. '''58''' (2009), 323-338.<br />
|- id="R"<br />
|[Ri96] || '''J. Rickard''', ''Splendid equivalences: derived categories and permutation modules'', Proc. London Math. Soc. '''72''' (1996), 331-358.<br />
|-<br />
|[Ro95] || '''R. Rouquier''', ''From stable equivalences to Rickard equivalences for blocks with cyclic defect'', Proceedings of Groups 1993, Galway-St. Andrews Conference, Vol. 2, London Math. Soc. Lecture Note Ser. '''212''', Cambridge University Press (1995), 512-523.<br />
|-<br />
|[Ru11] || '''P. Ruengrot''', ''Perfect isometry groups for blocks of finite groups'', PhD Thesis, University of Manchester (2011).<br />
|- id="S"<br />
|[Sa11] || '''B. Sambale''', ''<math>2</math>-blocks with minimal nonabelian defect groups'', J. Algebra '''337''' (2011), 261–284.<br />
|-<br />
|[Sa12] || '''B. Sambale''', ''Blocks with defect group <math>D_{2^n} \times C_{2^m}</math>'', J. Pure Appl. Algebra '''216''' (2012), 119–125.<br />
|-<br />
|[Sa12b] || '''B. Sambale''', ''Fusion systems on metacyclic 2-groups'', Osaka J. Math. '''49''' (2012), 325–329.<br />
|-<br />
|[Sa13] || '''B. Sambale''', ''Blocks with defect group <math>Q_{2^n} \times C_{2^m}</math> and <math>SD_{2^n} \times C_{2^m}</math>'', Algebr. Represent. Theory '''16''' (2013), 1717–1732.<br />
|-<br />
|[Sa13b] || '''B. Sambale''', ''Blocks with central product defect group <math>D_{2^n} ∗ C_{2^m}</math>'', Proc. Amer. Math. Soc. '''141''' (2013), 4057–4069.<br />
|-<br />
|[Sa13c] || '''B. Sambale''', ''Further evidence for conjectures in block theory'', Algebra Number Theory '''7''' (2013), 2241–2273. <br />
|-<br />
|[Sa14] || '''B. Sambale''', ''Blocks of Finite Groups and Their Invariants'', Lecture Notes in Mathematics, Springer (2014).<br />
|-<br />
|[Sa16] || '''B. Sambale''', ''2-blocks with minimal nonabelian defect groups III'', Pacific J. Math. '''280''' (2016), 475–487.<br />
|-<br />
|[Sa20] || '''B. Sambale''', [https://arxiv.org/abs/2005.13172 ''Blocks with small-dimensional basic algebra''], Bul. Aust. Math. Soc. '''103''' (2021), 461-474.<br />
|-<br />
|[SSS98] || '''M. Schaps, D. Shapira and O. Shlomo''', ''Quivers of blocks with normal defect groups'', Proc. Symp. in Pure Mathematics '''63''', Amer. Math. Soc. (1998), 497-510.<br />
|-<br />
|[Sc91] || '''J. Scopes''', ''Cartan matrices and Morita equivalence for blocks of the symmetric groups'', J. Algebra '''142''' (1991), 441-455.<br />
|-<br />
|[Sh20] || '''V. Shalotenko''', ''Bounds on the dimension of Ext for finite groups of Lie type'', J. Algebra '''550''' (2020), 266-289.<br />
|-<br />
|[St02] || '''R. Stancu''', ''Almost all generalized extraspecial p-groups are resistant'', J. Algebra '''249''' (2002), 120-126.<br />
|-<br />
|[St06] || '''R. Stancu''', ''Control of fusion in fusion systems'', J. Algebra and its Applications '''5''' (2006), 817-837. <br />
|- id="T"<br />
|[Th93] || '''J. Thévenaz''', ''Most finite groups are <math>p</math>-nilpotent'', Exposition. Math. '''11''' (1993), 359-363.<br />
|- id="V"<br />
|[vdW91] || '''R. van der Waall''', ''On p-nilpotent forcing groups'', Indag. Mathem., N.S., '''2''' (1991), 367-384.<br />
|- id="W"<br />
|[Wa00]|| '''A. Watanabe''', ''A remark on a splitting theorem for blocks with abelian defect groups'', RIMS Kokyuroku '''1140''', Edited by H.Sasaki, Research Institute for Mathematical Sciences, Kyoto University (2000), 76-79.<br />
|-<br />
|[WZZ18] || '''Chao Wu, Kun Zhang and Yuanyang Zhou''', ''[https://arxiv.org/abs/1803.07262 Blocks with defect group <math>Z_{2^n} \times Z_{2^n} \times Z_{2^m}</math>]'', J. Algebra '''510''' (2018), 469-498.<br />
|}</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Blocks_with_basic_algebras_of_low_dimension&diff=1229
Blocks with basic algebras of low dimension
2023-10-10T15:54:37Z
<p>Charles Eaton: </p>
<hr />
<div>== Blocks with basic algebras of dimension at most 16 ==<br />
<br />
In [[References#L|[Li18b]]] Markus Linckelmann calculated the <math>k</math>-algebras of dimension at most twelve which occur as basic algebras of blocks of finite groups, with the exception of one case of dimension 9 where no block with that basic algebra was identified<ref>The algebra of dimension 9 has the following structure.<br />
<br />
'''Quiver:''' a:<1,2>, b:<2,1>, c:<1,1>, d:<1,1><br />
<br />
'''Relations w.r.t. <math>k</math>:''' ab=c^3=d^2, cd=dc=0, ca=bc=da=bd=0<br />
<br />
'''Cartan matrix:''' <math>\left( \begin{array}{cc}<br />
5 & 1 \\<br />
1 & 2 \\<br />
\end{array} \right)</math><br />
<br />
A corresponding <math>\mathcal{O}</math>-block would have '''decomposition matrix''' <math>\left( \begin{array}{cc}<br />
1 & 0 \\<br />
1 & 0 \\<br />
1 & 0 \\<br />
1 & 0 \\<br />
0 & 1 \\<br />
1 & 1 \\<br />
\end{array}\right)</math><br />
<br />
Labelling the simple modules by <math>S_1, S_2</math>, the projective indecomposable modules have Loewy structure as follows:<br />
<br />
<math>\begin{array}{cc}<br />
\begin{array}{ccc}<br />
& S_1 & \\<br />
S_2 & \begin{array}{c} S_1 \\ S_1 \\ \end{array} & S_1 \\<br />
& S_1 & \\ <br />
\end{array} , & <br />
\begin{array}{c}<br />
S_2 \\<br />
S_1 \\<br />
S_2 \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
</ref>. This final case was ruled out by Linckelmann and Murphy in [[References#L|[LM20]]]. Using the classification of finite simple groups, the basic algebras of dimension 13 or 14 for blocks of finite groups were calculated by Sambale in [[References#S|[Sa20]]]. Later Benson and Sambale in [[References#B|[BS23]]] gave a classification for dimensions 15 and 16, except for one unsettled case of a block with defect group <math>C_{13}</math> in dimension 15.<br />
<br />
The results are incorporated into the table below. <br />
<br />
{| class="wikitable"<br />
|-<br />
! scope="col"| Dimension<br />
! scope="col"| Class<br />
! scope="col"| Defect group<br />
! scope="col"| Representative<br />
! scope="col"| <math>\dim_k(Z(A))</math><br />
! scope="col"| <math>l(A)</math><br />
! scope="col"| Notes<br />
|-<br />
| 1 || [[M(1,1,1)]] || <math>1</math> || <math>k1</math> || 1 || 1 || Blocks of defect zero<br />
|-<br />
| 2 || [[M(2,1,1)]] || <math>C_2</math> || <math>kC_2</math> || 2 || 1 || <br />
|-<br />
| 3 || [[M(3,1,1)]] || <math>C_3</math> || <math>kC_3</math> || 3 || 1 || <br />
|-<br />
| 4 || [[M(4,1,1)]] || <math>C_4</math> || <math>kC_4</math> || 4 || 1 ||<br />
|-<br />
| 4 || [[M(4,2,1)]] || <math>C_2 \times C_2</math> || <math>k(C_2 \times C_2)</math> || 4 || 1 ||<br />
|-<br />
| 5 || [[M(5,1,1)]] || <math>C_5</math> || <math>kC_5</math> || 5 || 1 ||<br />
|-<br />
| 6 || [[M(3,1,2)]] || <math>C_3</math> || <math>kS_3</math> || 3 || 2 ||<br />
|-<br />
| 7 || [[M(5,1,3)]] || <math>C_5</math> || <math>B_0(kA_5)</math> || 4 || 2 ||<br />
|-<br />
| 7 || [[M(7,1,1)]] || <math>C_7</math> || <math>kC_7</math> || 7 || 1 ||<br />
|-<br />
| 8 || [[M(8,1,1)]] || <math>C_8</math> || <math>kC_8</math> || 8 || 1 ||<br />
|-<br />
| 8 || [[M(8,2,1)]] || <math>C_4 \times C_2</math> || <math>k(C_4 \times C_2)</math> || 8 || 1 ||<br />
|-<br />
| 8 || [[M(8,3,1)]] || <math>D_8</math> || <math>kD_8</math> || 5 || 1 ||<br />
|-<br />
| 8 || [[M(8,4,1)]] || <math>Q_8</math> || <math>kQ_8</math> || 5 || 1 ||<br />
|-<br />
| 8 || [[M(8,5,1)]] || <math>C_2 \times C_2 \times C_2</math> || <math>k(C_2 \times C_2 \times C_2)</math> || 8 || 1 ||<br />
|-<br />
| 8 || [[M(7,1,3)]] || <math>C_7</math> || <math>B_0(kPSL_2(13))</math> || 5 || 2 ||<br />
|-<br />
| 9 || [[M(9,1,1)]] || <math>C_9</math> || <math>kC_9</math> || 9 || 1 ||<br />
|-<br />
| 9 || [[M(9,1,3)]] || <math>C_9</math> || <math>B_0(kSL_2(8))</math> || 6 || 2 ||<br />
|-<br />
| 9 || [[M(9,2,1)]] || <math>C_3 \times C_3</math> || <math>k(C_3 \times C_3)</math> || 9 || 1 ||<br />
|-<br />
| 9 || [[M(9,2,23)]] || <math>C_3 \times C_3</math> || Faithful block of <math>k((C_3 \times C_3):Q_8)</math>, in which <math>Z(Q_8)</math> acts trivially || 6 || 1 || SmallGroup(72,24)<br />
|-<br />
| 10 || [[M(5,1,2)]] || <math>C_5</math> || <math>kD_{10}</math> || 4 || 2 ||<br />
|-<br />
| 10 || [[M(11,1,3)]] || <math>C_{11}</math> || <math>B_0(kSL_2(32))</math> || 7 || 2 ||<br />
|-<br />
| 11 || [[M(8,3,3)]] || <math>D_8</math> || <math>kS_4</math> || 5 || 2 ||<br />
|-<br />
| 11 || [[M(7,1,6)]] || <math>C_7</math> || <math>B_0(kA_7)</math> || 5 || 3 ||<br />
|-<br />
| 11 || [[M(11,1,1)]] || <math>C_{11}</math> || <math>kC_{11}</math> || 11 || 1 ||<br />
|-<br />
| 11 || [[M(13,1,3)]] || <math>C_{13}</math> || <math>B_0(kPSL_2(25))</math> || 8 || 2 ||<br />
|-<br />
| 12 || [[M(4,2,3)]] || <math>C_2 \times C_2</math> || <math>kA_4</math> || 4 || 3 ||<br />
|-<br />
| 13 || [[M(16,7,3)]] || <math>D_{16}</math> || <math>B_0(kPGL_2(7))</math> || 7 || 2 ||<br />
|-<br />
| 13 || [[M(16,8,4)]] || <math>SD_{16}</math> || <math>B_3(k(3.M_{10}))</math> || 7 || 2 ||<br />
|-<br />
| 13 || [[M(7,1,7)]] || <math>C_7</math> || <math>B_{15}(k6.A_7)</math> || 5 || 3 ||<br />
|-<br />
| 13 || [[M(13,1,1)]] || <math>C_{13}</math> || <math>kC_{13}</math> || 13 || 1 ||<br />
|-<br />
| 13 || M(13,1,?) || <math>C_{13}</math> || <math>B_0(kPSL_3(3))</math> || 7 || 3 ||<br />
|-<br />
| 13 || M(17,1,?) || <math>C_{17}</math> || <math>B_0(kPSL_2(16))</math> || 10 || 2 ||<br />
|-<br />
| 14 || [[M(5,1,5)]] || <math>C_5</math> || <math>B_0(kS_5)</math> || 5 || 4 ||<br />
|-<br />
| 14 || [[M(7,1,2)]] || <math>C_7</math> || <math>kD_{14}</math> || 5 || 2 ||<br />
|-<br />
| 14 || [[M(7,1,5)]] || <math>C_7</math> || <math>B_0(kPSL_3(3))</math> || 5 || 3 ||<br />
|-<br />
| 14 || M(19,1,?) || <math>C_{19}</math> || <math>B_0(kPSL_2(37))</math> || 11 || 2 ||<br />
|-<br />
| 15 || M(19,1,?) || <math>C_{19}</math> || <math>B_0(kGL_3(7))</math> || ||<br />
|-<br />
| 15 || || <math>C_{13}</math> || ?? || || <br />
<br />
|}<br />
<br />
== Notes ==<br />
<br />
<references /></div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=References&diff=1228
References
2023-10-10T15:51:38Z
<p>Charles Eaton: [BS23]</p>
<hr />
<div>[[#A|A,]] [[#B|B,]] [[#C|C,]] [[#D|D,]] [[#E|E,]] [[#F|F,]] [[#G|G,]] [[#H|H,]] [[#I|I,]] [[#J|J,]] [[#K|K,]] [[#L|L,]] [[#M|M,]] [[#N|N,]] [[#O|O,]] [[#P|P,]] [[#Q|Q,]] [[#R|R,]] [[#S|S,]] [[#T|T]] [[#U|U,]] [[#V|V,]] [[#W|W,]] [[#X|X,]] [[#Y|Y,]] [[#Z|Z,]] <br />
<br />
{| <br />
|- id="A"<br />
|[Al79] || '''J. L. Alperin''', ''Projective modules for <math>SL(2,2^n)</math>'', J. Pure and Applied Algebra '''15''' (1979), 219-234.<br />
|-<br />
|[Al80] || '''J. L. Alperin''', ''Local representation theory'', The Santa Cruz Conference on Finite Groups., Proc. Sympos. Pure Math. '''37''' (1980), 369-375.<br />
|-<br />
|[AE81] || '''J. L. Alperin and L. Evens''', ''Representations, resoluutions and Quillen's dimension theorem'', J. Pure Appl. Algebra '''22''' (1981), 1-9.<br />
|-<br />
|[AE04] || '''Jianbei An and C. W. Eaton''', ''Blocks with trivial intersection defect groups'', Math. Z. '''247''' (2004), 461-486.<br />
|-<br />
|[AE04] || '''Jianbei An and C. W. Eaton''', [https://arxiv.org/abs/2310.02150 ''Morita equivalence classes of blocks with extraspecial defect groups <math>p_+^{1+2}</math>''], [https://arxiv.org/abs/2310.02150 arxiv:2310.02150]<br />
|-<br />
|[Ar19] || '''C. G. Ardito''', [https://arxiv.org/abs/1908.02652 ''Morita equivalence classes of blocks with elementary abelian defect groups of order 32''], J. Algebra '''573''' (2021), 297-335.<br />
|-<br />
|[ArMcK20] || '''C. G. Ardito and E. McKernon''', ''[https://arxiv.org/abs/2010.08329 ''2-blocks with an abelian defect group and a freely acting cyclic inertial quotient''], [https://arxiv.org/abs/2010.08329 arxiv.org/abs/2010.08329]<br />
|-<br />
|[AS20] || '''C. G. Ardito and B. Sambale''', [http://www.advgrouptheory.com/journal/Volumes/12/ArditoSambale.pdf ''Broué's Conjecture for 2-blocks with elementary abelian defect groups of order 32''], Advances in Group Theory and Applications 12 (2021), 71–78. <br />
|-<br />
|[AKO11] || '''M. Aschbacher, R. Kessar and B. Oliver''', ''Fusion systems in algebra and topology'', London Mathematical Society Lecture Notes '''391''', Cambridge University Press (2011).<br />
|- id="B"<br />
|[BK07] || '''D. Benson and R. Kessar''', ''Blocks inequivalent to their Frobenius twists'', J. Algebra '''315''' (2007), 588-599.<br />
|-<br />
|[BS23] || '''D. Benson and B. Sambale''', [https://arxiv.org/abs/2301.10537 ''Finite dimensional algebras not arising as blocks in group algebras''], [https://arxiv.org/pdf/2301.10537 arxiv:2301.10537]<br />
|-<br />
|[BKL18] || '''R. Boltje, R. Kessar, and M. Linckelmann''', [https://doi.org/10.1016/j.jalgebra.2019.02.045 ''On Picard groups of blocks of finite groups''], J. Algebra '''558''' (2020), 70-101.<br />
|-<br />
|[Bra41] || '''R. Brauer''', ''Investigations on group characters'', Ann. Math. '''42''' (1941), 936-958.<br />
|-<br />
|[BP80] || '''M. Broué and L. Puig''', ''A Frobenius theorem for blocks'', Invent. Math. '''56''' (1980), 117-128.<br />
|-<br />
|[BP80b] || '''M. Broué and L. Puig''', ''Characters and local structure in G-algebras'', J. Algebra '''63''' (1980), 306-317.<br />
|- id="C"<br />
|[Cr11] || '''D. A. Craven''', ''The Theory of Fusion Systems: An Algebraic Approach'', Cambridge University Press (2011).<br />
|-<br />
|[Cr12] || '''D. A. Craven''', [https://arxiv.org/abs/1207.0116 ''Perverse Equivalences and Broué's Conjecture II: The Cyclic Case''], [https://arxiv.org/abs/1207.0116 arXiv:1207.0116]<br />
|-<br />
|[CDR18] || '''D. A. Craven, O. Dudas and R. Rouquier''', [https://arxiv.org/abs/1701.07097 ''The Brauer trees of unipotent blocks''], to appear, J. EMS, [https://arxiv.org/abs/1701.07097 arXiv:1701.07097] <br />
|-<br />
|[CEKL11] || '''D. A. Craven, C. W. Eaton, R. Kessar and M. Linckelmann''', ''The structure of blocks with a Klein four defect group'', Math. Z. '''268''' (2011), 441-476.<br />
|-<br />
|[CG12] || '''D. A. Craven and A. Glesser''', ''Fusion systems on small p-groups'', Trans. AMS '''364''' (2012) 5945-5967.<br />
|-<br />
|[CR13] || '''D. A. Craven and R. Rouquier''', ''Perverse equivalences and Broué's conjecture'', Adv. Math. '''248''' (2013), 1-58.<br />
|-<br />
|[CuRe81a] || '''C. W. Curtis and I. Reiner''', ''Methods of representation theory, with applications to finite groups and orders, Volume I'', Wiley-Interscience (1981).<br />
|-<br />
|[CuRe81b] || '''C. W. Curtis and I. Reiner''', ''Methods of representation theory, with applications to finite groups and orders, Volume II'', Wiley-Interscience (1981).<br />
|- id="D"<br />
|[Da66] || '''E. C. Dade''', ''Blocks with cyclic defect groups'', Ann. Math. '''84''' (1966), 20-48. <br />
|-<br />
|[DE20] || '''S. Danz and K. Erdmann''', [https://arxiv.org/abs/2008.10999 ''On Ext-Quivers of Blocks of weight two for symmetric groups''], [https://arxiv.org/abs/2008.10999 arXiv:2008.10999]<br />
|-<br />
|[Du14] || '''O. Dudas''', [https://arxiv.org/abs/1011.5478 ''Coxeter orbits and Brauer trees II''], Int. Math. Res. Not. '''15''' (2014), 4100-4123.<br />
|-<br />
|[Dü04] || '''O. Düvel''', ''On Donovan's conjecture'', J. Algebra '''272''' (2004), 1-26.<br />
|- id="E"<br />
|[Ea16] || '''C. W. Eaton''', ''Morita equivalence classes of <math>2</math>-blocks of defect three'', Proc. AMS '''144''' (2016), 1961-1970.<br />
|-<br />
|[Ea18] || '''C. W. Eaton''', [https://arxiv.org/abs/1612.03485 ''Morita equivalence classes of blocks with elementary abelian defect groups of order 16''], [https://arxiv.org/abs/1612.03485 arXiv:1612.03485]<br />
|-<br />
|[EEL18] || '''C. W. Eaton, F. Eisele and M. Livesey''', [https://arxiv.org/abs/1809.08152 ''Donovan's conjecture, blocks with abelian defect groups and discrete valuation rings''], Math. Z. '''295''' (2020), 249-264.<br />
|-<br />
|[EKKS14] || '''C. W. Eaton, R. Kessar, B. Külshammer and B. Sambale''', ''<math>2</math>-blocks with abelian defect groups'', Adv. Math. '''254''' (2014), 706-735.<br />
|-<br />
|[EKS12] || '''C. W. Eaton, B. Külshammer and B. Sambale''', ''<math>2</math>-blocks with minimal nonabelian defect groups, II'', J. Group Theory '''15''' (2012), 311-321.<br />
|-<br />
|[EL18a] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/1709.04331 Classifying blocks with abelian defect groups of rank 3 for the prime 2]'', J. Algebra '''515''' (2018), 1-18.<br />
|-<br />
|[EL18b] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/1803.03539 Donovan's conjecture and blocks with abelian defect groups]'', Proc. AMS. '''147''' (2019), 963-970.<br />
|-<br />
|[EL18c] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/1810.10950 Some examples of Picard groups of blocks]'', J. Algebra '''558''' (2020), 350-370.<br />
|-<br />
|[EL20] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/2006.11173 Donovan's conjecture and extensions by the centralizer of a defect group]'', J. Algebra '''582''' (2021), 157-176.<br />
|-<br />
|[EL23] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/2310.05734 Morita equivalence classes of <math>2</math>-blocks with abelian defect groups of rank <math>4</math>]'', [https://arxiv.org/abs/2310.05734 arxiv:2310.05734]<br />
|-<br />
|[Ei16] || '''F. Eisele''', ''Blocks with a generalized quaternion defect group and three simple modules over a <math>2</math>-adic ring'', J. Algebra '''456''' (2016), 294-322.<br />
|-<br />
|[Ei18] || '''F. Eisele''', ''[https://arxiv.org/abs/1807.05110 The Picard group of an order and Külshammer reduction]'', Algebr. Represent. Theory '''24''' (2021), 505-518.<br />
|-<br />
|[Ei19] || '''F. Eisele''', ''[https://arxiv.org/abs/1908.00129 On the geometry of lattices and finiteness of Picard groups]'', J. Reine Angew. Math. '''782''' (2022), 219-333. <br />
|-<br />
|[EiLiv20] || '''F. Eisele and M. Livesey''', ''[https://arxiv.org/abs/2006.13837 Arbitrarily large Morita Frobenius numbers]'', Algebra Number Theory '''16''' (2022), 1889-1904.<br />
|-<br />
|[Er82] || '''K. Erdmann''', ''Blocks whose defect groups are Klein four groups: a correction'', J. Algebra '''76''' (1982), 505-518.<br />
|-<br />
|[Er87] || '''K. Erdmann''', ''Algebras and dihedral defect groups'', Proc. LMS '''54''' (1987), 88-114.<br />
|-<br />
|[Er88a] || '''K. Erdmann''', ''Algebras and quaternion defect groups, I'', Math. Ann. '''281''' (1988), 545-560.<br />
|-<br />
|[Er88b] || '''K. Erdmann''', ''Algebras and quaternion defect groups, II'', Math. Ann. '''281''' (1988), 561-582. <br />
|-<br />
|[Er88c] || '''K. Erdmann''', ''Algebras and semidihedral defect groups I'', Proc. LMS '''57''' (1988), 109-150. <br />
|-<br />
|[Er90] || ''' K. Erdmann''', ''Blocks of tame representation type and related algebras'', Lecture Notes in Mathematics '''1428''', Springer-Verlag (1990).<br />
|-<br />
|[Er90b] || '''K. Erdmann''', ''Algebras and semidihedral defect groups II'', Proc. LMS '''60''' (1990), 123-165.<br />
|- id="F"<br />
|[Fa17] || '''N. Farrell''', ''On the Morita Frobenius numbers of blocks of finite reductive groups'', J. Algebra '''471''' (2017), 299-318.<br />
|-<br />
|[FK18] || '''N. Farrell and R. Kessar''', [https://arxiv.org/abs/1805.02015 ''Rationality of blocks of quasi-simple finite groups''], Represent. Theory '''23''' (2019), 325-349.<br />
|- id="G"<br />
|[GMdelR21] || '''D. Garcia, l. Margolis and A. del Rio''', [https://arxiv.org/abs/2016.07231 ''Non-isomorphic 2-groups with isomorphic modular group algebras''], J. Reine Angew. Math. '''f783''' (2022), 269–274.<br />
|-<br />
|[GO97] || '''H. Gollan and T. Okuyama''', ''Derived equivalences for the smallest Janko group'', preprint (1997).<br />
|-<br />
|[GT19] || '''R. M. Guralnick and Pham Huu Tiep''', ''Sectional rank and Cohomology'', J. Algebra (2019) https://doi.org/10.1016/j.jalgebra.2019.04.023<br />
|- id="H"<br />
|[HM07] || '''G. T. Helleloid and U. Martin''', ''The automorphism group of a finite <math>p</math>-group is almost always a <math>p</math>-group'', J. Algebra (2007), 294-329.<br />
|-<br />
|[HP94] || '''H-W. Henn and S. Priddy''', ''<math>p</math>-nilpotence, classifying space indecompsability, and other properties of almost finite groups'', Comment. Math. Helvetici (1994), 335-350.<br />
|- <br />
|[HK00] || '''G. Hiss and R. Kessar''', ''Scopes reduction and Morita equivalence classes of blocks in finite classical groups'', J. Algebra '''230''' (2000), 378-423.<br />
|-<br />
|[HK05] || '''G. Hiss and R. Kessar''', ''Scopes reduction and Morita equivalence classes of blocks in finite classical groups II'', J. Algebra '''283''' (2005), 522-563.<br />
|-<br />
|[Ho97] || '''T. Holm''', ''Derived equivalent tame blocks'', J. Algebra '''194''' (1997), 178-200.<br />
|-<br />
|[HKL07] || '''T. Holm, R. Kessar and M. Linckelmann''', ''Blocks with a quaternion defect group over a 2-adic ring: the case <math>\tilde{A}_4</math>'', Glasgow Math. J. '''49''' (2007), 29–43.<br />
|- id="J"<br />
|[Ja69] || '''G. Janusz''', ''Indecomposable modules for finite groups'', Ann. Math. '''89''' (1969), 209-241.<br />
|-<br />
|[Jo96] || '''T. Jost''', ''Morita equivalences for blocks of finite general linear groups'', Manuscripta Math. '''91''' (1996), 121-144.<br />
|- id="K"<br />
|[Ke96] || '''R. Kessar''', ''Blocks and source algebras for the double covers of the symmetric and alternating groups'', J. Algebra '''186''' (1996), 872-933.<br />
|-<br />
|[Ke00] || '''R. Kessar''', ''Equivalences for blocks of the Weyl groups'', Proc. Amer. Math. Soc. '''128''' (2000), 337-346.<br />
|-<br />
|[Ke01] || '''R. Kessar''', ''Source algebra equivalences for blocks of finite general linear groups over a fixed field'', Manuscripta Math. '''104''' (2001), 145-162. <br />
|-<br />
|[Ke02] || '''R. Kessar''', ''Scopes reduction for blocks of finite alternating groups'', Quart. J. Math. '''53''' (2002), 443-454.<br />
|-<br />
|[Ke05] || ''' R. Kessar''', ''A remark on Donovan's conjecture'', Arch. Math (Basel) '''82''' (2005), 391-394.<br />
|-<br />
|[KL18] || '''R. Kessar and M. Linckelmann''', [https://arxiv.org/abs/1705.07227 ''Descent of equivalences and character bijections''], [https://arxiv.org/abs/1705.07227 arXiv:1705.07227]<br />
|-<br />
|[Ki84] || '''M. Kiyota''', ''On 3-blocks with an elementary abelian defect group of order 9'', J. Fac. Sci. Univ. Tokyo Sect. IA Math. '''31''' (1984), 33–58.<br />
|-<br />
|[Ko03] || '''S. Koshitani''', ''Conjectures of Donovan and Puig for principal <math>3</math>-blocks with abelian defect groups'', Comm. Alg. '''31''' (2003), 2229-2243; ''Corrigendum'', '''32''' (2004), 391-393.<br />
|-<br />
|[KKW02] || '''S. Koshitani, N. Kunugi and K. Waki''', ''Broué's conjecture for non-principal 3-blocks of finite groups'', J. Pure and Applied Algebra '''173''' (2002), 177-211. <br />
|-<br />
|[KKW04] || '''S. Koshitani, N. Kunugi and K. Waki''', ''Broué's abelian defect group conjecture for Held group and the sporadic Suzuki group'', J. Algebra '''279''' (2004), 638-666. <br />
|-<br />
|[KoLa20] || '''S. Koshitani and C. Lassueur''', ''Splendid Morita equivalences for principal 2-blocks with dihedral defect groups'', Math. Z. '''294''' (2020), 639-666.<br />
|-<br />
|[KoLa20b] || '''S. Koshitani and C. Lassueur''', ''Splendid Morita equivalences for principal blocks with generalised quaternion defect groups'', J. Algebra '''558''' (2020), 523-533.<br />
|-<br />
|[KoLaSa22] || '''S. Koshitani, C. Lassueur and B. Sambale''', ''Splendid Morita equivalences for principal blocks with semidihedral defect groups'', Proceedings of the American Mathematical Society '''150''' (2022), 41-53.<br />
|-<br />
|[Kü80] || '''B. Külshammer''', ''On 2-blocks with wreathed defect groups'', J. Algebra '''64''' (1980), 529–555.<br />
|-<br />
|[Kü81] || '''B. Külshammer''', ''On p-blocks of p-solvable groups'', Comm. Alg. '''9''' (1981), 1763-1785.<br />
|-<br />
|[Kü91] || '''B. Külshammer''', ''Group-theoretical descriptions of ring-theoretical invariants of group algebras'', in Representation Theory of Finite Groups and Finite-Dimensional Algebras (Bielefeld, 1991), Progr. Math. '''95''', pp. 425-442, Birkhauser (1991).<br />
|-<br />
|[Kü95] || '''B. Külshammer''', ''Donovan's conjecture, crossed products and algebraic group actions'', Israel J. Math. '''92''' (1995), 295-306.<br />
|-<br />
|[KS13] || '''B. Külshammer and B. Sambale''', ''The 2-blocks of defect 4'', Representation Theory '''17''' (2013), 226-236.<br />
|-<br />
|[Ku00] || '''N. Kunugi''', ''Morita equivalent 3-blocks of the 3-dimensional projective special linear groups'', Proc. LMS '''80''' (2000), 575-589.<br />
|-<br />
|[Kup69] || '''H. Kupisch''', ''Unzerlegbare Moduln endlicher Gruppen mit zyklischer p-Sylow Gruppe'', Math. Z. '''108''' (1969), 77-104.<br />
|- id="L"<br />
|[LM80]||'''P. Landrock and G. O. Michler''', ''Principal 2-blocks of the simple groups of Ree type'', Trans. AMS '''260''' (1980), 83-111.<br />
|-<br />
|[Li94] || '''M. Linckelmann''', ''The source algebras of blocks with a Klein four defect group'', J. Algebra '''167''' (1994), 821-854.<br />
|-<br />
|[Li94b] || '''M. Linckelmann''', ''A derived equivalence for blocks with dihedral defect groups'', J. Algebra '''164''' (1994), 244-255. <br />
|-<br />
|[Li96] || '''M. Linckelmann''', ''The isomorphism problem for cyclic blocks and their source algebras'', Invent. Math. '''125''' (1996), 265-283.<br />
|-<br />
|[Li18] || '''M. Linckelmann''', [https://arxiv.org/abs/1805.08884 ''The strong Frobenius numbers for cyclic defect blocks are equal to one''], [https://arxiv.org/abs/1805.08884 arXiv:1805.08884]<br />
|-<br />
|[Li18b] || '''M. Linckelmann''', ''Finite-dimensional algebras arising as blocks of finite group algebras'', Contemporary Mathematics '''705''' (2018), 155-188.<br />
|-<br />
|[Li18c] || '''M. Linckelmann''', ''The block theory of finite group algebras, Volume 1'', London Math. Soc. Student Texts '''92''', Cambridge University Press (2018).<br />
|-<br />
|[Li18d] || '''M. Linckelmann''', ''The block theory of finite group algebras, Volume 2'', London Math. Soc. Student Texts '''92''', Cambridge University Press (2018).<br />
|-<br />
|[LM20] || '''M. Linckelmann and W. Murphy''', [https://arxiv.org/abs/2005.02223 ''A 9-dimensional algebra which is not a block of a finite group''], Quarterly Journal of Mathematics 72 (2021), 1077–1088<br />
|-<br />
|[Liv19] || '''M. Livesey''', [https://arxiv.org/abs/1907.12167 ''On Picard groups of blocks with normal defect groups''], J. Algebra '''566''' (2021), 94-118.<br />
|-<br />
|[LiMa20] || '''M. Livesey and C. Marchi''', [https://arxiv.org/abs/2002.10571 ''On Picent for blocks with normal defect group''], [https://arxiv.org/abs/2002.10571 arXiv:2002.10571]<br />
|-<br />
|[LiMa20b] || '''M. Livesey and C. Marchi''', [https://arxiv.org/abs/2008.05857 ''Picard groups for blocks with normal defect groups and linear source bimodules''], [https://arxiv.org/abs/2008.05857 arXiv:2008.05857]<br />
|- id="M"<br />
|[Mac] || '''N. Macgregor''', ''Morita equivalence classes of tame blocks of finite groups'', J. Algebra '''608''' (2022), 719-754.<br />
|-<br />
|[Mar] || '''C. Marchi''', ''Picard groups for blocks'', PhD thesis, University of Manchester (2022)<br />
|-<br />
|[Ma86] || '''U. Martin''', ''Almost all <math>p</math>-groups have automorphism group a <math>p</math>-group'', Bull. AMS '''15''' (1986), 78-82.<br />
|-<br />
|[McK19] || '''E. McKernon''', [https://arxiv.org/abs/1912.03222 ''2-Blocks whose defect group is homocyclic and whose inertial quotient contains a Singer cycle''], J. Algebra '''563''' (2020), 30–48.<br />
|-<br />
|[MS08] || '''J. Müller and M. Schaps''', ''The Broué conjecture for the faithful 3-blocks of <math>4.M_{22}</math>'', J. Algebra '''319''' (2008), 3588-3602.<br />
|- id="N"<br />
|[NS18] || '''G. Navarro and B. Sambale''', ''On the blockwise modular isomorphism problem'', Manuscripta Math. '''157''' (2018), 263-278.<br />
|- <br />
|[Ne02] || '''G. Nebe''', [http://www.math.rwth-aachen.de/~Gabriele.Nebe/papers/survey.pdf ''Group rings of finite groups over p-adic integers, some examples''], Proceedings of the conference Around Group rings (Edmonton) Resenhas '''5''' (2002), 329-350.<br />
|- id="O"<br />
|[Ok97] || '''T. Okuyama''', ''Some examples of derived equivalent blocks of finite groups'', preprint (1997).<br />
|- id="P"<br />
|[Pu88]|| '''L. Puig''', ''Nilpotent blocks and their source algebras'', Invent. Math. '''93''' (1988), 77-116.<br />
|-<br />
|[Pu94] || '''L. Puig''', ''On Joanna Scopes’ criterion of equivalence for blocks of symmetric groups'', Algebra Colloq. '''1''' (1994), 25-55.<br />
|-<br />
|[Pu99]|| '''L. Puig''', ''On the local structure of Morita and Rickard equivalences between Brauer blocks'', Progress in Math. '''178''', Birkhauser Verlag (1999).<br />
|-<br />
|[Pu09] || '''L. Puig''', ''Block source algebras in p-solvable groups'', Michigan Math. J. '''58''' (2009), 323-338.<br />
|- id="R"<br />
|[Ri96] || '''J. Rickard''', ''Splendid equivalences: derived categories and permutation modules'', Proc. London Math. Soc. '''72''' (1996), 331-358.<br />
|-<br />
|[Ro95] || '''R. Rouquier''', ''From stable equivalences to Rickard equivalences for blocks with cyclic defect'', Proceedings of Groups 1993, Galway-St. Andrews Conference, Vol. 2, London Math. Soc. Lecture Note Ser. '''212''', Cambridge University Press (1995), 512-523.<br />
|-<br />
|[Ru11] || '''P. Ruengrot''', ''Perfect isometry groups for blocks of finite groups'', PhD Thesis, University of Manchester (2011).<br />
|- id="S"<br />
|[Sa11] || '''B. Sambale''', ''<math>2</math>-blocks with minimal nonabelian defect groups'', J. Algebra '''337''' (2011), 261–284.<br />
|-<br />
|[Sa12] || '''B. Sambale''', ''Blocks with defect group <math>D_{2^n} \times C_{2^m}</math>'', J. Pure Appl. Algebra '''216''' (2012), 119–125.<br />
|-<br />
|[Sa12b] || '''B. Sambale''', ''Fusion systems on metacyclic 2-groups'', Osaka J. Math. '''49''' (2012), 325–329.<br />
|-<br />
|[Sa13] || '''B. Sambale''', ''Blocks with defect group <math>Q_{2^n} \times C_{2^m}</math> and <math>SD_{2^n} \times C_{2^m}</math>'', Algebr. Represent. Theory '''16''' (2013), 1717–1732.<br />
|-<br />
|[Sa13b] || '''B. Sambale''', ''Blocks with central product defect group <math>D_{2^n} ∗ C_{2^m}</math>'', Proc. Amer. Math. Soc. '''141''' (2013), 4057–4069.<br />
|-<br />
|[Sa13c] || '''B. Sambale''', ''Further evidence for conjectures in block theory'', Algebra Number Theory '''7''' (2013), 2241–2273. <br />
|-<br />
|[Sa14] || '''B. Sambale''', ''Blocks of Finite Groups and Their Invariants'', Lecture Notes in Mathematics, Springer (2014).<br />
|-<br />
|[Sa16] || '''B. Sambale''', ''2-blocks with minimal nonabelian defect groups III'', Pacific J. Math. '''280''' (2016), 475–487.<br />
|-<br />
|[Sa20] || '''B. Sambale''', [https://arxiv.org/abs/2005.13172 ''Blocks with small-dimensional basic algebra''], Bul. Aust. Math. Soc. '''103''' (2021), 461-474.<br />
|-<br />
|[SSS98] || '''M. Schaps, D. Shapira and O. Shlomo''', ''Quivers of blocks with normal defect groups'', Proc. Symp. in Pure Mathematics '''63''', Amer. Math. Soc. (1998), 497-510.<br />
|-<br />
|[Sc91] || '''J. Scopes''', ''Cartan matrices and Morita equivalence for blocks of the symmetric groups'', J. Algebra '''142''' (1991), 441-455.<br />
|-<br />
|[Sh20] || '''V. Shalotenko''', ''Bounds on the dimension of Ext for finite groups of Lie type'', J. Algebra '''550''' (2020), 266-289.<br />
|-<br />
|[St02] || '''R. Stancu''', ''Almost all generalized extraspecial p-groups are resistant'', J. Algebra '''249''' (2002), 120-126.<br />
|-<br />
|[St06] || '''R. Stancu''', ''Control of fusion in fusion systems'', J. Algebra and its Applications '''5''' (2006), 817-837. <br />
|- id="T"<br />
|[Th93] || '''J. Thévenaz''', ''Most finite groups are <math>p</math>-nilpotent'', Exposition. Math. '''11''' (1993), 359-363.<br />
|- id="V"<br />
|[vdW91] || '''R. van der Waall''', ''On p-nilpotent forcing groups'', Indag. Mathem., N.S., '''2''' (1991), 367-384.<br />
|- id="W"<br />
|[Wa00]|| '''A. Watanabe''', ''A remark on a splitting theorem for blocks with abelian defect groups'', RIMS Kokyuroku '''1140''', Edited by H.Sasaki, Research Institute for Mathematical Sciences, Kyoto University (2000), 76-79.<br />
|-<br />
|[WZZ18] || '''Chao Wu, Kun Zhang and Yuanyang Zhou''', ''[https://arxiv.org/abs/1803.07262 Blocks with defect group <math>Z_{2^n} \times Z_{2^n} \times Z_{2^m}</math>]'', J. Algebra '''510''' (2018), 469-498.<br />
|}</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Blocks_with_basic_algebras_of_low_dimension&diff=1227
Blocks with basic algebras of low dimension
2023-10-10T15:46:09Z
<p>Charles Eaton: /* Blocks with basic algebras of dimension at most 14 */</p>
<hr />
<div>== Blocks with basic algebras of dimension at most 14 ==<br />
<br />
In [[References#L|[Li18b]]] Markus Linckelmann calculated the <math>k</math>-algebras of dimension at most twelve which occur as basic algebras of blocks of finite groups, with the exception of one case of dimension 9 where no block with that basic algebra was identified<ref>The algebra of dimension 9 has the following structure.<br />
<br />
'''Quiver:''' a:<1,2>, b:<2,1>, c:<1,1>, d:<1,1><br />
<br />
'''Relations w.r.t. <math>k</math>:''' ab=c^3=d^2, cd=dc=0, ca=bc=da=bd=0<br />
<br />
'''Cartan matrix:''' <math>\left( \begin{array}{cc}<br />
5 & 1 \\<br />
1 & 2 \\<br />
\end{array} \right)</math><br />
<br />
A corresponding <math>\mathcal{O}</math>-block would have '''decomposition matrix''' <math>\left( \begin{array}{cc}<br />
1 & 0 \\<br />
1 & 0 \\<br />
1 & 0 \\<br />
1 & 0 \\<br />
0 & 1 \\<br />
1 & 1 \\<br />
\end{array}\right)</math><br />
<br />
Labelling the simple modules by <math>S_1, S_2</math>, the projective indecomposable modules have Loewy structure as follows:<br />
<br />
<math>\begin{array}{cc}<br />
\begin{array}{ccc}<br />
& S_1 & \\<br />
S_2 & \begin{array}{c} S_1 \\ S_1 \\ \end{array} & S_1 \\<br />
& S_1 & \\ <br />
\end{array} , & <br />
\begin{array}{c}<br />
S_2 \\<br />
S_1 \\<br />
S_2 \\<br />
\end{array}<br />
\end{array}<br />
</math><br />
</ref>. This final case was ruled out by Linckelmann and Murphy in [[References#L|[LM20]]]. Using the classification of finite simple groups, the basic algebras of dimension 13 or 14 for blocks of finite groups were calculated by Sambale in [[References#S|[Sa20]]].<br />
<br />
The results are incorporated into the table below. <br />
<br />
{| class="wikitable"<br />
|-<br />
! scope="col"| Dimension<br />
! scope="col"| Class<br />
! scope="col"| Defect group<br />
! scope="col"| Representative<br />
! scope="col"| <math>\dim_k(Z(A))</math><br />
! scope="col"| <math>l(A)</math><br />
! scope="col"| Notes<br />
|-<br />
| 1 || [[M(1,1,1)]] || <math>1</math> || <math>k1</math> || 1 || 1 || Blocks of defect zero<br />
|-<br />
| 2 || [[M(2,1,1)]] || <math>C_2</math> || <math>kC_2</math> || 2 || 1 || <br />
|-<br />
| 3 || [[M(3,1,1)]] || <math>C_3</math> || <math>kC_3</math> || 3 || 1 || <br />
|-<br />
| 4 || [[M(4,1,1)]] || <math>C_4</math> || <math>kC_4</math> || 4 || 1 ||<br />
|-<br />
| 4 || [[M(4,2,1)]] || <math>C_2 \times C_2</math> || <math>k(C_2 \times C_2)</math> || 4 || 1 ||<br />
|-<br />
| 5 || [[M(5,1,1)]] || <math>C_5</math> || <math>kC_5</math> || 5 || 1 ||<br />
|-<br />
| 6 || [[M(3,1,2)]] || <math>C_3</math> || <math>kS_3</math> || 3 || 2 ||<br />
|-<br />
| 7 || [[M(5,1,3)]] || <math>C_5</math> || <math>B_0(kA_5)</math> || 4 || 2 ||<br />
|-<br />
| 7 || [[M(7,1,1)]] || <math>C_7</math> || <math>kC_7</math> || 7 || 1 ||<br />
|-<br />
| 8 || [[M(8,1,1)]] || <math>C_8</math> || <math>kC_8</math> || 8 || 1 ||<br />
|-<br />
| 8 || [[M(8,2,1)]] || <math>C_4 \times C_2</math> || <math>k(C_4 \times C_2)</math> || 8 || 1 ||<br />
|-<br />
| 8 || [[M(8,3,1)]] || <math>D_8</math> || <math>kD_8</math> || 5 || 1 ||<br />
|-<br />
| 8 || [[M(8,4,1)]] || <math>Q_8</math> || <math>kQ_8</math> || 5 || 1 ||<br />
|-<br />
| 8 || [[M(8,5,1)]] || <math>C_2 \times C_2 \times C_2</math> || <math>k(C_2 \times C_2 \times C_2)</math> || 8 || 1 ||<br />
|-<br />
| 8 || [[M(7,1,3)]] || <math>C_7</math> || <math>B_0(kPSL_2(13))</math> || 5 || 2 ||<br />
|-<br />
| 9 || [[M(9,1,1)]] || <math>C_9</math> || <math>kC_9</math> || 9 || 1 ||<br />
|-<br />
| 9 || [[M(9,1,3)]] || <math>C_9</math> || <math>B_0(kSL_2(8))</math> || 6 || 2 ||<br />
|-<br />
| 9 || [[M(9,2,1)]] || <math>C_3 \times C_3</math> || <math>k(C_3 \times C_3)</math> || 9 || 1 ||<br />
|-<br />
| 9 || [[M(9,2,23)]] || <math>C_3 \times C_3</math> || Faithful block of <math>k((C_3 \times C_3):Q_8)</math>, in which <math>Z(Q_8)</math> acts trivially || 6 || 1 || SmallGroup(72,24)<br />
|-<br />
| 10 || [[M(5,1,2)]] || <math>C_5</math> || <math>kD_{10}</math> || 4 || 2 ||<br />
|-<br />
| 10 || [[M(11,1,3)]] || <math>C_{11}</math> || <math>B_0(kSL_2(32))</math> || 7 || 2 ||<br />
|-<br />
| 11 || [[M(8,3,3)]] || <math>D_8</math> || <math>kS_4</math> || 5 || 2 ||<br />
|-<br />
| 11 || [[M(7,1,6)]] || <math>C_7</math> || <math>B_0(kA_7)</math> || 5 || 3 ||<br />
|-<br />
| 11 || [[M(11,1,1)]] || <math>C_{11}</math> || <math>kC_{11}</math> || 11 || 1 ||<br />
|-<br />
| 11 || [[M(13,1,3)]] || <math>C_{13}</math> || <math>B_0(kPSL_2(25))</math> || 8 || 2 ||<br />
|-<br />
| 12 || [[M(4,2,3)]] || <math>C_2 \times C_2</math> || <math>kA_4</math> || 4 || 3 ||<br />
|-<br />
| 13 || [[M(16,7,3)]] || <math>D_{16}</math> || <math>B_0(kPGL_2(7))</math> || 7 || 2 ||<br />
|-<br />
| 13 || [[M(16,8,4)]] || <math>SD_{16}</math> || <math>B_3(k(3.M_{10}))</math> || 7 || 2 ||<br />
|-<br />
| 13 || [[M(7,1,7)]] || <math>C_7</math> || <math>B_{15}(k6.A_7)</math> || 5 || 3 ||<br />
|-<br />
| 13 || [[M(13,1,1)]] || <math>C_{13}</math> || <math>kC_{13}</math> || 13 || 1 ||<br />
|-<br />
| 13 || M(13,1,?) || <math>C_{13}</math> || <math>B_0(kPSL_3(3))</math> || 7 || 3 ||<br />
|-<br />
| 13 || M(17,1,?) || <math>C_{17}</math> || <math>B_0(kPSL_2(16))</math> || 10 || 2 ||<br />
|-<br />
| 14 || [[M(5,1,5)]] || <math>C_5</math> || <math>B_0(kS_5)</math> || 5 || 4 ||<br />
|-<br />
| 14 || [[M(7,1,2)]] || <math>C_7</math> || <math>kD_{14}</math> || 5 || 2 ||<br />
|-<br />
| 14 || [[M(7,1,5)]] || <math>C_7</math> || <math>B_0(kPSL_3(3))</math> || 5 || 3 ||<br />
|-<br />
| 14 || M(19,1,?) || <math>C_{19}</math> || <math>B_0(kPSL_2(37))</math> || 11 || 2 ||<br />
|-<br />
| 15 || M(19,1,?) || <math>C_{19}</math> || <math>B_0(kGL_3(7))</math> || ||<br />
|-<br />
| 15 || || <math>C_{13}</math> || ?? || || <br />
<br />
|}<br />
<br />
== Notes ==<br />
<br />
<references /></div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Generic_classifications_by_p-group_class&diff=1226
Generic classifications by p-group class
2023-10-10T14:36:28Z
<p>Charles Eaton: /* Abelian 2-groups with 2-rank at most four */</p>
<hr />
<div>This page contains results for classes of ''p''-groups for which we either have classifications or have general results concerning Morita equivalence classes. <br />
<br />
== Fusion trivial ''p''-groups ==<br />
<br />
''p''-groups <math>P</math> for which the only saturated fusion system is <math>\mathcal{F}_P(P)</math> have not yet been given a name in the literature (to our knowledge). We will call them ''fusion trivial'', but ''nilpotent forcing'' also seems appropriate following [[References#V|[vdW91]]] (where ''p''-groups for which any finite group <math>G</math> containing <math>P</math> as a Sylow ''p''-subgroup must be ''p''-nilpotent are called ''p-nilpotent forcing''). It is not known whether these two definitions are equivalent, i.e., whether there exist ''p''-nilpotent forcing ''p''-groups for which there is an exotic fusion system.<br />
<br />
Blocks with fusion trivial defect groups must be nilpotent and so Morita equivalent to the group algebra of a defect group by [[References#P|[Pu88]]].<br />
<br />
Examples of fusion trivial ''p''-groups are abelian ''2''-groups with automorphism group a ''2''-group (i.e., those whose cyclic factors have pairwise distinct orders), and metacyclic 2-groups other than homocyclic, dihedral, generalised quaternion or semidihedral groups (see [[References#C|[CG12]]] or [[References#S|[Sa12b]]]).<br />
<br />
Note that a ''p''-group is fusion trivial if and only if it is resistant and has automorphism group a ''p''-group. See [[References#S|[St06]]] for an analysis of resistant ''p''-groups.<br />
<br />
== Cyclic ''p''-groups ==<br />
<br />
[[Blocks with cyclic defect groups|Click here for background on blocks with cyclic defect groups]].<br />
<br />
Morita equivalence classes are labelled by [[Brauer trees]], but it is at present an open problem as to which Brauer trees are realised by blocks of finite groups. Each ''k''-Morita equivalence class corresponds to an unique <math>\mathcal{O}</math>-Morita equivalence class. <br />
<br />
For <math>p=2,3</math> every appropriate Brauer tree is realised by a block and we can give generic descriptions.<br />
<br />
[[C(2^n)|<math>2</math>-blocks with cyclic defect groups]]<br />
<br />
[[C(3^n)|<math>3</math>-blocks with cyclic defect groups]]<br />
<br />
== Tame blocks ==<br />
<br />
[[Tame blocks|Click here for background on tame blocks]].<br />
<br />
Erdmann classified algebras which are candidates for [[Basic algebras|basic algebras]] of tame blocks, i.e., those with dihedral, semidihedral or generalised quaternion defect groups (see [[References|[Er90] ]]) and in the cases of dihedral and semihedral defect groups determined which are realised by blocks of finite groups. In the case of generalised quaternion groups, the case of blocks with two simple modules is still open. These classifications only hold with respect to the field ''k'' at present.<br />
<br />
Principal blocks with dihedral defect groups are classified up to source algebra equivalence in [[References#K|[KoLa20]]].<br />
<br />
== Abelian <math>2</math>-groups with <math>2</math>-rank at most four ==<br />
<br />
[[Image:under-construction.png|50px|left]]<br />
<br />
These have been classified in [[References#W|[WZZ18]]], [[References#E|[EL18a]]] and [[References#E|[EL23]]] with respect to <math>\mathcal{O}</math>. The derived equivalences classes with respect to <math>\mathcal{O}</math> are known.<br />
<br />
Let <math>l,m,n,r \geq 2</math> be distinct.<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>rk_2(D) \leq 4</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>D</math> <br />
! scope="col"| # <math>(\mathcal{O}</math>-)Morita classes<br />
! scope="col"| # <math>\mathcal{O}</math>-Derived equiv classes<br />
! scope="col"| Always endopermutation source Morita equivalent?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|[[C(2^n)|<math>C_{2^n}</math>]] || 1 || 1 || Yes || [[References#L|[Li96]]] || [[Blocks with cyclic defect groups]]<br />
|-<br />
|[[C2xC2|<math>C_2 \times C_2</math>]] || 3 || 2 || Yes || [[References#L|[Li94]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)|<math>C_{2^m} \times C_{2^m}</math>]] || 2 || 2 || Yes || [[References#E|[EKKS14]]] ||<br />
|-<br />
|[[C(2^m)xC(2^n)|<math>C_{2^m} \times C_{2^n}</math>]] || 1 || 1 || Yes || [[References#P|[Pu88]]] || [[Nilpotent blocks]]<br />
|-<br />
|[[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]] || 8 || 4 || || [[References#E|[Ea16]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)xC(2^m)|<math>C_{2^m} \times C_{2^m} \times C_{2^m}</math>]] || 4 || 4 || Yes || [[References#W|[WZZ18]]] ||<br />
|-<br />
|[[C(2^m)xC2xC2|<math>C_{2^m} \times C_2 \times C_2</math>]] || 3 || 2 || || [[References#E|[EL18a]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)xC(2^n)|<math>C_{2^m} \times C_{2^m} \times C_{2^n}</math>]] || 2 || 2 || || [[References#W|[WZZ18]]] ||<br />
|-<br />
|[[C(2^l)xC(2^m)xC(2^n)|<math>C_{2^l} \times C_{2^m} \times C_{2^n}</math>]] || 1 || 1 || Yes || [[References#P|[Pu88]]] || [[Nilpotent blocks]]<br />
|-<br />
|[[(C2)^4|<math>(C_2)^4</math>]] || 16 || || || [[References#E|[Ea18]]] ||<br />
|-<br />
|[[C(2^m)xC2xC2xC2|<math>C_{2^m} \times C_2 \times C_2 \times C_2</math>]] || || || || [[References#E|[EL23]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)xC2xC2|<math>C_{2^m} \times C_{2^m} \times C_2 \times C_2</math>]] || || || || [[References#E|[EL23]]] ||<br />
|-<br />
|[[C(2^m)xC(2^n)xC2xC2|<math>C_{2^m} \times C_{2^n} \times C_2 \times C_2</math>]] || || || || [[References#E|[EL23]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)xC(2^n)xC(2^n)|<math>C_{2^m} \times C_{2^m} \times C_{2^n} \times C_{2^n}</math>]] || || || || [[References#E|[EL23]]] ||<br />
|-<br />
|[[C(2^l)xC(2^l)xC(2^m)xC(2^n)|<math>C_{2^l} \times C_{2^l} \times C_{2^m} \times C_{2^n}</math>]] || || || || [[References#E|[EL23]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)xC(2^m)xC(2^n)|<math>C_{2^m} \times C_{2^m} \times C_{2^m} \times C_{2^n}</math>]] || || || || [[References#E|[EL23]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)xC(2^m)xC(2^m)|<math>C_{2^m} \times C_{2^m} \times C_{2^m} \times C_{2^m}</math>]] || || || || [[References#E|[EL23]]] ||<br />
|-<br />
|[[C(2^l)xC(2^m)xC(2^n)xC(2^r)|<math>C_{2^l} \times C_{2^m} \times C_{2^n} \times C_{2^n}</math>]] || || || || [[References#E|[EL23]]] || [[Nilpotent blocks]] <br />
|}<br />
<br />
<br />
<!--== Abelian <math>2</math>-groups ==<br />
<br />
Donovan's conjecture holds for ''2''-blocks with abelian defect groups. Some generic classification results are known for certain inertial quotients. These will be detailed here.--><br />
<br />
== Minimal nonabelian <math>2</math>-groups ==<br />
<br />
Blocks with defect groups which are minimal nonabelian <math>2</math>-groups of the form <math>P=\langle x,y:x^{2^r}=y^{2^r}=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle</math> are classified in [[References#E|[EKS12]]]. There are two <math>\mathcal{O}</math>-Morita equivalence classes, with representatives <math>\mathcal{O}P</math> and <math>\mathcal{O}(P:C_3)</math>.<br />
<br />
For arbitrary minimal nonabelian <math>2</math>-groups, by [[References#S|[Sa16]]] blocks with such defect groups and the same fusion system are isotypic.<br />
<br />
== Homocyclic <math>2</math>-groups when inertial quotient contains a Singer cycle == <br />
<br />
A Singer cycle is an element of order <math>p^n-1</math> in <math>\operatorname{Aut}({C_p}^n) \cong GL_n(p)</math>, and a subgroup generated by a Singer cycle acts transitively on the non-trivial elements of <math>(C_p)^n</math>.<br />
<br />
<math>2</math>-blocks with homocyclic defect group <math> D \cong (C_{2^m})^n </math> whose inertial quotient <math> \mathbb{E} </math> contains a Singer cycle are classified in [[References#E|[McK19]]]. In this situation, <math>\mathbb{E}</math> has the form <math>E:F </math> where <math>E \cong C_{2^n-1}</math> and <math>F</math> is trivial or a subgroup of <math>C_n</math>. There are three <math> \mathcal{O} </math>-Morita equivalence classes when <math>m=1, n =3 </math>; two when <math>m=1 , n \neq 3 </math>; and only one when <math>m> 1 </math>. The three classes have representatives [[M(8,5,7) | <math> \mathcal{O}J_1 </math>]], which occurs only when <math>m=1, n=3</math>; <math> \mathcal{O}(SL_2(2^n):F) </math>, which occurs only when <math>m=1 </math>; and <math> \mathcal{O}(D : \mathbb{E})</math>. <br />
<br />
The Morita equivalence between the block and the class representative is known to be basic, possibly except when <math>m=1,n=3</math>, since the Morita equivalences between the [[M(8,5,8) |principal block of <math>{\rm \operatorname{Aut}(SL_2(8))}</math>]] and the blocks [[M(8,5,8)#Other_notatable_representatives |<math>B_0(\mathcal{O}({}^2G_2(3^{2m+1})))</math>]] are not known to be basic.<br />
<br />
<br />
== Abelian <math>2</math>-groups when inertial quotient is cyclic and acts freely on the defect group == <br />
<br />
<math>2</math>-blocks with abelian defect group <math> D </math> whose inertial quotient <math> E </math> is a cyclic group that acts freely on the defect group (i.e. such that the stabiliser in <math> E </math> of any nontrivial element of <math>D</math> is trivial) are classified in [[References#E|[ArMcK20]]].<br />
<br />
If the action of <math>E</math> is transitive, then <math>D</math> is homocyclic, <math>E</math> is a Singer cycle and, by the previous section, the block is either Morita equivalent to the principal block of <math> \mathcal{O}SL_2(2^n) </math>, or to <math> \mathcal{O}(D : E)</math>. <br />
<br />
If the action of <math>E</math> is not transitive, then the block is Morita equivalent to <math> \mathcal{O}(D : E)</math>. <br />
<br />
In each case, the Morita equivalence between the block and the class representative is known to be basic.<br />
<br />
== Extraspecial <math>p</math>-groups of order <math>p^3</math> and exponent <math>p</math> for <math>p \geq 5</math> ==<br />
<br />
These blocks are described in [[References#A|[AE23]]]. Donovan's conjecture holds in this case.</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Generic_classifications_by_p-group_class&diff=1225
Generic classifications by p-group class
2023-10-10T12:49:55Z
<p>Charles Eaton: /* Abelian 2-groups with 2-rank at most four */</p>
<hr />
<div>This page contains results for classes of ''p''-groups for which we either have classifications or have general results concerning Morita equivalence classes. <br />
<br />
== Fusion trivial ''p''-groups ==<br />
<br />
''p''-groups <math>P</math> for which the only saturated fusion system is <math>\mathcal{F}_P(P)</math> have not yet been given a name in the literature (to our knowledge). We will call them ''fusion trivial'', but ''nilpotent forcing'' also seems appropriate following [[References#V|[vdW91]]] (where ''p''-groups for which any finite group <math>G</math> containing <math>P</math> as a Sylow ''p''-subgroup must be ''p''-nilpotent are called ''p-nilpotent forcing''). It is not known whether these two definitions are equivalent, i.e., whether there exist ''p''-nilpotent forcing ''p''-groups for which there is an exotic fusion system.<br />
<br />
Blocks with fusion trivial defect groups must be nilpotent and so Morita equivalent to the group algebra of a defect group by [[References#P|[Pu88]]].<br />
<br />
Examples of fusion trivial ''p''-groups are abelian ''2''-groups with automorphism group a ''2''-group (i.e., those whose cyclic factors have pairwise distinct orders), and metacyclic 2-groups other than homocyclic, dihedral, generalised quaternion or semidihedral groups (see [[References#C|[CG12]]] or [[References#S|[Sa12b]]]).<br />
<br />
Note that a ''p''-group is fusion trivial if and only if it is resistant and has automorphism group a ''p''-group. See [[References#S|[St06]]] for an analysis of resistant ''p''-groups.<br />
<br />
== Cyclic ''p''-groups ==<br />
<br />
[[Blocks with cyclic defect groups|Click here for background on blocks with cyclic defect groups]].<br />
<br />
Morita equivalence classes are labelled by [[Brauer trees]], but it is at present an open problem as to which Brauer trees are realised by blocks of finite groups. Each ''k''-Morita equivalence class corresponds to an unique <math>\mathcal{O}</math>-Morita equivalence class. <br />
<br />
For <math>p=2,3</math> every appropriate Brauer tree is realised by a block and we can give generic descriptions.<br />
<br />
[[C(2^n)|<math>2</math>-blocks with cyclic defect groups]]<br />
<br />
[[C(3^n)|<math>3</math>-blocks with cyclic defect groups]]<br />
<br />
== Tame blocks ==<br />
<br />
[[Tame blocks|Click here for background on tame blocks]].<br />
<br />
Erdmann classified algebras which are candidates for [[Basic algebras|basic algebras]] of tame blocks, i.e., those with dihedral, semidihedral or generalised quaternion defect groups (see [[References|[Er90] ]]) and in the cases of dihedral and semihedral defect groups determined which are realised by blocks of finite groups. In the case of generalised quaternion groups, the case of blocks with two simple modules is still open. These classifications only hold with respect to the field ''k'' at present.<br />
<br />
Principal blocks with dihedral defect groups are classified up to source algebra equivalence in [[References#K|[KoLa20]]].<br />
<br />
== Abelian <math>2</math>-groups with <math>2</math>-rank at most four ==<br />
<br />
[[Image:under-construction.png|50px|left]]<br />
<br />
These have been classified in [[References#W|[WZZ18]]], [[References#E|[EL18a]]] and [[References#E|[EL23]]] with respect to <math>\mathcal{O}</math>. The derived equivalences classes with respect to <math>\mathcal{O}</math> are known.<br />
<br />
Let <math>l,m,n,r \geq 2</math> be distinct.<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>rk_2(D) \leq 4</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>D</math> <br />
! scope="col"| # <math>(\mathcal{O}</math>-)Morita classes<br />
! scope="col"| # <math>\mathcal{O}</math>-Derived equiv classes<br />
! scope="col"| Always endopermutation source Morita equivalent?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|[[C(2^n)|<math>C_{2^n}</math>]] || 1 || 1 || Yes || [[References#L|[Li96]]] || [[Blocks with cyclic defect groups]]<br />
|-<br />
|[[C2xC2|<math>C_2 \times C_2</math>]] || 3 || 2 || Yes || [[References#L|[Li94]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)|<math>C_{2^m} \times C_{2^m}</math>]] || 2 || 2 || Yes || [[References#E|[EKKS14]]] ||<br />
|-<br />
|[[C(2^m)xC(2^n)|<math>C_{2^m} \times C_{2^n}</math>]] || 1 || 1 || Yes || [[References#P|[Pu88]]] || [[Nilpotent blocks]]<br />
|-<br />
|[[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]] || 8 || 4 || || [[References#E|[Ea16]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)xC(2^m)|<math>C_{2^m} \times C_{2^m} \times C_{2^m}</math>]] || 4 || 4 || Yes || [[References#W|[WZZ18]]] ||<br />
|-<br />
|[[C(2^m)xC2xC2|<math>C_{2^m} \times C_2 \times C_2</math>]] || 3 || 2 || || [[References#E|[EL18a]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)xC(2^n)|<math>C_{2^m} \times C_{2^m} \times C_{2^n}</math>]] || 2 || 2 || || [[References#W|[WZZ18]]] ||<br />
|-<br />
|[[C(2^l)xC(2^m)xC(2^n)|<math>C_{2^l} \times C_{2^m} \times C_{2^n}</math>]] || 1 || 1 || Yes || [[References#P|[Pu88]]] || [[Nilpotent blocks]]<br />
|-<br />
|[[(C2)^4|<math>(C_2)^4</math>]] || 16 || || || [[References#E|[Ea18]]] ||<br />
|-<br />
|[[C(2^m)xC2xC2xC2|<math>C_{2^m} \times C_2 \times C_2 \times C_2</math>]] || || || || [[References#E|[EL23]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)xC2xC2|<math>C_{2^m} \times C_{2^m} \times C_2 \times C_2</math>]] || || || || [[References#E|[EL23]]] ||<br />
|}<br />
<br />
<br />
<!--== Abelian <math>2</math>-groups ==<br />
<br />
Donovan's conjecture holds for ''2''-blocks with abelian defect groups. Some generic classification results are known for certain inertial quotients. These will be detailed here.--><br />
<br />
== Minimal nonabelian <math>2</math>-groups ==<br />
<br />
Blocks with defect groups which are minimal nonabelian <math>2</math>-groups of the form <math>P=\langle x,y:x^{2^r}=y^{2^r}=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle</math> are classified in [[References#E|[EKS12]]]. There are two <math>\mathcal{O}</math>-Morita equivalence classes, with representatives <math>\mathcal{O}P</math> and <math>\mathcal{O}(P:C_3)</math>.<br />
<br />
For arbitrary minimal nonabelian <math>2</math>-groups, by [[References#S|[Sa16]]] blocks with such defect groups and the same fusion system are isotypic.<br />
<br />
== Homocyclic <math>2</math>-groups when inertial quotient contains a Singer cycle == <br />
<br />
A Singer cycle is an element of order <math>p^n-1</math> in <math>\operatorname{Aut}({C_p}^n) \cong GL_n(p)</math>, and a subgroup generated by a Singer cycle acts transitively on the non-trivial elements of <math>(C_p)^n</math>.<br />
<br />
<math>2</math>-blocks with homocyclic defect group <math> D \cong (C_{2^m})^n </math> whose inertial quotient <math> \mathbb{E} </math> contains a Singer cycle are classified in [[References#E|[McK19]]]. In this situation, <math>\mathbb{E}</math> has the form <math>E:F </math> where <math>E \cong C_{2^n-1}</math> and <math>F</math> is trivial or a subgroup of <math>C_n</math>. There are three <math> \mathcal{O} </math>-Morita equivalence classes when <math>m=1, n =3 </math>; two when <math>m=1 , n \neq 3 </math>; and only one when <math>m> 1 </math>. The three classes have representatives [[M(8,5,7) | <math> \mathcal{O}J_1 </math>]], which occurs only when <math>m=1, n=3</math>; <math> \mathcal{O}(SL_2(2^n):F) </math>, which occurs only when <math>m=1 </math>; and <math> \mathcal{O}(D : \mathbb{E})</math>. <br />
<br />
The Morita equivalence between the block and the class representative is known to be basic, possibly except when <math>m=1,n=3</math>, since the Morita equivalences between the [[M(8,5,8) |principal block of <math>{\rm \operatorname{Aut}(SL_2(8))}</math>]] and the blocks [[M(8,5,8)#Other_notatable_representatives |<math>B_0(\mathcal{O}({}^2G_2(3^{2m+1})))</math>]] are not known to be basic.<br />
<br />
<br />
== Abelian <math>2</math>-groups when inertial quotient is cyclic and acts freely on the defect group == <br />
<br />
<math>2</math>-blocks with abelian defect group <math> D </math> whose inertial quotient <math> E </math> is a cyclic group that acts freely on the defect group (i.e. such that the stabiliser in <math> E </math> of any nontrivial element of <math>D</math> is trivial) are classified in [[References#E|[ArMcK20]]].<br />
<br />
If the action of <math>E</math> is transitive, then <math>D</math> is homocyclic, <math>E</math> is a Singer cycle and, by the previous section, the block is either Morita equivalent to the principal block of <math> \mathcal{O}SL_2(2^n) </math>, or to <math> \mathcal{O}(D : E)</math>. <br />
<br />
If the action of <math>E</math> is not transitive, then the block is Morita equivalent to <math> \mathcal{O}(D : E)</math>. <br />
<br />
In each case, the Morita equivalence between the block and the class representative is known to be basic.<br />
<br />
== Extraspecial <math>p</math>-groups of order <math>p^3</math> and exponent <math>p</math> for <math>p \geq 5</math> ==<br />
<br />
These blocks are described in [[References#A|[AE23]]]. Donovan's conjecture holds in this case.</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Generic_classifications_by_p-group_class&diff=1224
Generic classifications by p-group class
2023-10-10T08:22:58Z
<p>Charles Eaton: /* Abelian 2-groups with 2-rank at most three */</p>
<hr />
<div>This page contains results for classes of ''p''-groups for which we either have classifications or have general results concerning Morita equivalence classes. <br />
<br />
== Fusion trivial ''p''-groups ==<br />
<br />
''p''-groups <math>P</math> for which the only saturated fusion system is <math>\mathcal{F}_P(P)</math> have not yet been given a name in the literature (to our knowledge). We will call them ''fusion trivial'', but ''nilpotent forcing'' also seems appropriate following [[References#V|[vdW91]]] (where ''p''-groups for which any finite group <math>G</math> containing <math>P</math> as a Sylow ''p''-subgroup must be ''p''-nilpotent are called ''p-nilpotent forcing''). It is not known whether these two definitions are equivalent, i.e., whether there exist ''p''-nilpotent forcing ''p''-groups for which there is an exotic fusion system.<br />
<br />
Blocks with fusion trivial defect groups must be nilpotent and so Morita equivalent to the group algebra of a defect group by [[References#P|[Pu88]]].<br />
<br />
Examples of fusion trivial ''p''-groups are abelian ''2''-groups with automorphism group a ''2''-group (i.e., those whose cyclic factors have pairwise distinct orders), and metacyclic 2-groups other than homocyclic, dihedral, generalised quaternion or semidihedral groups (see [[References#C|[CG12]]] or [[References#S|[Sa12b]]]).<br />
<br />
Note that a ''p''-group is fusion trivial if and only if it is resistant and has automorphism group a ''p''-group. See [[References#S|[St06]]] for an analysis of resistant ''p''-groups.<br />
<br />
== Cyclic ''p''-groups ==<br />
<br />
[[Blocks with cyclic defect groups|Click here for background on blocks with cyclic defect groups]].<br />
<br />
Morita equivalence classes are labelled by [[Brauer trees]], but it is at present an open problem as to which Brauer trees are realised by blocks of finite groups. Each ''k''-Morita equivalence class corresponds to an unique <math>\mathcal{O}</math>-Morita equivalence class. <br />
<br />
For <math>p=2,3</math> every appropriate Brauer tree is realised by a block and we can give generic descriptions.<br />
<br />
[[C(2^n)|<math>2</math>-blocks with cyclic defect groups]]<br />
<br />
[[C(3^n)|<math>3</math>-blocks with cyclic defect groups]]<br />
<br />
== Tame blocks ==<br />
<br />
[[Tame blocks|Click here for background on tame blocks]].<br />
<br />
Erdmann classified algebras which are candidates for [[Basic algebras|basic algebras]] of tame blocks, i.e., those with dihedral, semidihedral or generalised quaternion defect groups (see [[References|[Er90] ]]) and in the cases of dihedral and semihedral defect groups determined which are realised by blocks of finite groups. In the case of generalised quaternion groups, the case of blocks with two simple modules is still open. These classifications only hold with respect to the field ''k'' at present.<br />
<br />
Principal blocks with dihedral defect groups are classified up to source algebra equivalence in [[References#K|[KoLa20]]].<br />
<br />
== Abelian <math>2</math>-groups with <math>2</math>-rank at most four ==<br />
<br />
These have been classified in [[References#W|[WZZ18]]], [[References#E|[EL18a]]] and [[References#E|[EL23]]] with respect to <math>\mathcal{O}</math>. The derived equivalences classes with respect to <math>\mathcal{O}</math> are known.<br />
<br />
Let <math>l,m,n \geq 1</math> be distinct with <math>l,m \neq 1</math><br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>rk_2(D) \leq 3</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>D</math> <br />
! scope="col"| # <math>(\mathcal{O}</math>-)Morita classes<br />
! scope="col"| # <math>\mathcal{O}</math>-Derived equiv classes<br />
! scope="col"| Always endopermutation source Morita equivalent?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|[[C(2^n)|<math>C_{2^n}</math>]] || 1 || 1 || Yes || [[References#L|[Li96]]] || [[Blocks with cyclic defect groups]]<br />
|-<br />
|[[C2xC2|<math>C_2 \times C_2</math>]] || 3 || 2 || Yes || [[References#L|[Li94]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)|<math>C_{2^m} \times C_{2^m}</math>]] || 2 || 2 || Yes || [[References#E|[EKKS14]]] ||<br />
|-<br />
|[[C(2^m)xC(2^n)|<math>C_{2^m} \times C_{2^n}</math>]] || 1 || 1 || Yes || [[References#P|[Pu88]]] || [[Nilpotent blocks]]<br />
|-<br />
|[[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]] || 8 || 4 || || [[References#E|[Ea16]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)xC(2^m)|<math>C_{2^m} \times C_{2^m} \times C_{2^m}</math>]] || 4 || 4 || Yes || [[References#W|[WZZ18]]] ||<br />
|-<br />
|[[C(2^m)xC2xC2|<math>C_{2^m} \times C_2 \times C_2</math>]] || 3 || 2 || || [[References#E|[EL18a]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)xC(2^n)|<math>C_{2^m} \times C_{2^m} \times C_{2^n}</math>]] || 2 || 2 || || [[References#W|[WZZ18]]] ||<br />
|-<br />
|[[C(2^l)xC(2^m)xC(2^n)|<math>C_{2^l} \times C_{2^m} \times C_{2^n}</math>]] || 1 || 1 || Yes || [[References#P|[Pu88]]] || [[Nilpotent blocks]]<br />
|}<br />
<br />
<!--== Abelian <math>2</math>-groups ==<br />
<br />
Donovan's conjecture holds for ''2''-blocks with abelian defect groups. Some generic classification results are known for certain inertial quotients. These will be detailed here.--><br />
<br />
== Minimal nonabelian <math>2</math>-groups ==<br />
<br />
Blocks with defect groups which are minimal nonabelian <math>2</math>-groups of the form <math>P=\langle x,y:x^{2^r}=y^{2^r}=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle</math> are classified in [[References#E|[EKS12]]]. There are two <math>\mathcal{O}</math>-Morita equivalence classes, with representatives <math>\mathcal{O}P</math> and <math>\mathcal{O}(P:C_3)</math>.<br />
<br />
For arbitrary minimal nonabelian <math>2</math>-groups, by [[References#S|[Sa16]]] blocks with such defect groups and the same fusion system are isotypic.<br />
<br />
== Homocyclic <math>2</math>-groups when inertial quotient contains a Singer cycle == <br />
<br />
A Singer cycle is an element of order <math>p^n-1</math> in <math>\operatorname{Aut}({C_p}^n) \cong GL_n(p)</math>, and a subgroup generated by a Singer cycle acts transitively on the non-trivial elements of <math>(C_p)^n</math>.<br />
<br />
<math>2</math>-blocks with homocyclic defect group <math> D \cong (C_{2^m})^n </math> whose inertial quotient <math> \mathbb{E} </math> contains a Singer cycle are classified in [[References#E|[McK19]]]. In this situation, <math>\mathbb{E}</math> has the form <math>E:F </math> where <math>E \cong C_{2^n-1}</math> and <math>F</math> is trivial or a subgroup of <math>C_n</math>. There are three <math> \mathcal{O} </math>-Morita equivalence classes when <math>m=1, n =3 </math>; two when <math>m=1 , n \neq 3 </math>; and only one when <math>m> 1 </math>. The three classes have representatives [[M(8,5,7) | <math> \mathcal{O}J_1 </math>]], which occurs only when <math>m=1, n=3</math>; <math> \mathcal{O}(SL_2(2^n):F) </math>, which occurs only when <math>m=1 </math>; and <math> \mathcal{O}(D : \mathbb{E})</math>. <br />
<br />
The Morita equivalence between the block and the class representative is known to be basic, possibly except when <math>m=1,n=3</math>, since the Morita equivalences between the [[M(8,5,8) |principal block of <math>{\rm \operatorname{Aut}(SL_2(8))}</math>]] and the blocks [[M(8,5,8)#Other_notatable_representatives |<math>B_0(\mathcal{O}({}^2G_2(3^{2m+1})))</math>]] are not known to be basic.<br />
<br />
<br />
== Abelian <math>2</math>-groups when inertial quotient is cyclic and acts freely on the defect group == <br />
<br />
<math>2</math>-blocks with abelian defect group <math> D </math> whose inertial quotient <math> E </math> is a cyclic group that acts freely on the defect group (i.e. such that the stabiliser in <math> E </math> of any nontrivial element of <math>D</math> is trivial) are classified in [[References#E|[ArMcK20]]].<br />
<br />
If the action of <math>E</math> is transitive, then <math>D</math> is homocyclic, <math>E</math> is a Singer cycle and, by the previous section, the block is either Morita equivalent to the principal block of <math> \mathcal{O}SL_2(2^n) </math>, or to <math> \mathcal{O}(D : E)</math>. <br />
<br />
If the action of <math>E</math> is not transitive, then the block is Morita equivalent to <math> \mathcal{O}(D : E)</math>. <br />
<br />
In each case, the Morita equivalence between the block and the class representative is known to be basic.<br />
<br />
== Extraspecial <math>p</math>-groups of order <math>p^3</math> and exponent <math>p</math> for <math>p \geq 5</math> ==<br />
<br />
These blocks are described in [[References#A|[AE23]]]. Donovan's conjecture holds in this case.</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=References&diff=1223
References
2023-10-10T08:20:52Z
<p>Charles Eaton: [EL23] and updated [Ei19], [EiLiv20]</p>
<hr />
<div>[[#A|A,]] [[#B|B,]] [[#C|C,]] [[#D|D,]] [[#E|E,]] [[#F|F,]] [[#G|G,]] [[#H|H,]] [[#I|I,]] [[#J|J,]] [[#K|K,]] [[#L|L,]] [[#M|M,]] [[#N|N,]] [[#O|O,]] [[#P|P,]] [[#Q|Q,]] [[#R|R,]] [[#S|S,]] [[#T|T]] [[#U|U,]] [[#V|V,]] [[#W|W,]] [[#X|X,]] [[#Y|Y,]] [[#Z|Z,]] <br />
<br />
{| <br />
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|-<br />
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|-<br />
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|-<br />
|[AE04] || '''Jianbei An and C. W. Eaton''', ''Blocks with trivial intersection defect groups'', Math. Z. '''247''' (2004), 461-486.<br />
|-<br />
|[AE04] || '''Jianbei An and C. W. Eaton''', ''[https://arxiv.org/abs/2310.02150 "Morita equivalence classes of blocks with extraspecial defect groups <math>p_+^{1+2}</math>''], [https://arxiv.org/abs/2310.02150 arxiv.org/abs/2310.02150]<br />
|-<br />
|[Ar19] || '''C. G. Ardito''', [https://arxiv.org/abs/1908.02652 ''Morita equivalence classes of blocks with elementary abelian defect groups of order 32''], J. Algebra '''573''' (2021), 297-335.<br />
|-<br />
|[ArMcK20] || '''C. G. Ardito and E. McKernon''', ''[https://arxiv.org/abs/2010.08329 ''2-blocks with an abelian defect group and a freely acting cyclic inertial quotient''], [https://arxiv.org/abs/2010.08329 arxiv.org/abs/2010.08329]<br />
|-<br />
|[AS20] || '''C. G. Ardito and B. Sambale''', [http://www.advgrouptheory.com/journal/Volumes/12/ArditoSambale.pdf ''Broué's Conjecture for 2-blocks with elementary abelian defect groups of order 32''], Advances in Group Theory and Applications 12 (2021), 71–78. <br />
|-<br />
|[AKO11] || '''M. Aschbacher, R. Kessar and B. Oliver''', ''Fusion systems in algebra and topology'', London Mathematical Society Lecture Notes '''391''', Cambridge University Press (2011).<br />
|- id="B"<br />
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|-<br />
|[BKL18] || '''R. Boltje, R. Kessar, and M. Linckelmann''', [https://doi.org/10.1016/j.jalgebra.2019.02.045 ''On Picard groups of blocks of finite groups''], J. Algebra '''558''' (2020), 70-101.<br />
|-<br />
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|-<br />
|[CDR18] || '''D. A. Craven, O. Dudas and R. Rouquier''', [https://arxiv.org/abs/1701.07097 ''The Brauer trees of unipotent blocks''], to appear, J. EMS, [https://arxiv.org/abs/1701.07097 arXiv:1701.07097] <br />
|-<br />
|[CEKL11] || '''D. A. Craven, C. W. Eaton, R. Kessar and M. Linckelmann''', ''The structure of blocks with a Klein four defect group'', Math. Z. '''268''' (2011), 441-476.<br />
|-<br />
|[CG12] || '''D. A. Craven and A. Glesser''', ''Fusion systems on small p-groups'', Trans. AMS '''364''' (2012) 5945-5967.<br />
|-<br />
|[CR13] || '''D. A. Craven and R. Rouquier''', ''Perverse equivalences and Broué's conjecture'', Adv. Math. '''248''' (2013), 1-58.<br />
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|[CuRe81a] || '''C. W. Curtis and I. Reiner''', ''Methods of representation theory, with applications to finite groups and orders, Volume I'', Wiley-Interscience (1981).<br />
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|[CuRe81b] || '''C. W. Curtis and I. Reiner''', ''Methods of representation theory, with applications to finite groups and orders, Volume II'', Wiley-Interscience (1981).<br />
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|[Da66] || '''E. C. Dade''', ''Blocks with cyclic defect groups'', Ann. Math. '''84''' (1966), 20-48. <br />
|-<br />
|[DE20] || '''S. Danz and K. Erdmann''', [https://arxiv.org/abs/2008.10999 ''On Ext-Quivers of Blocks of weight two for symmetric groups''], [https://arxiv.org/abs/2008.10999 arXiv:2008.10999]<br />
|-<br />
|[Du14] || '''O. Dudas''', [https://arxiv.org/abs/1011.5478 ''Coxeter orbits and Brauer trees II''], Int. Math. Res. Not. '''15''' (2014), 4100-4123.<br />
|-<br />
|[Dü04] || '''O. Düvel''', ''On Donovan's conjecture'', J. Algebra '''272''' (2004), 1-26.<br />
|- id="E"<br />
|[Ea16] || '''C. W. Eaton''', ''Morita equivalence classes of <math>2</math>-blocks of defect three'', Proc. AMS '''144''' (2016), 1961-1970.<br />
|-<br />
|[Ea18] || '''C. W. Eaton''', [https://arxiv.org/abs/1612.03485 ''Morita equivalence classes of blocks with elementary abelian defect groups of order 16''], [https://arxiv.org/abs/1612.03485 arXiv:1612.03485]<br />
|-<br />
|[EEL18] || '''C. W. Eaton, F. Eisele and M. Livesey''', [https://arxiv.org/abs/1809.08152 ''Donovan's conjecture, blocks with abelian defect groups and discrete valuation rings''], Math. Z. '''295''' (2020), 249-264.<br />
|-<br />
|[EKKS14] || '''C. W. Eaton, R. Kessar, B. Külshammer and B. Sambale''', ''<math>2</math>-blocks with abelian defect groups'', Adv. Math. '''254''' (2014), 706-735.<br />
|-<br />
|[EKS12] || '''C. W. Eaton, B. Külshammer and B. Sambale''', ''<math>2</math>-blocks with minimal nonabelian defect groups, II'', J. Group Theory '''15''' (2012), 311-321.<br />
|-<br />
|[EL18a] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/1709.04331 Classifying blocks with abelian defect groups of rank 3 for the prime 2]'', J. Algebra '''515''' (2018), 1-18.<br />
|-<br />
|[EL18b] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/1803.03539 Donovan's conjecture and blocks with abelian defect groups]'', Proc. AMS. '''147''' (2019), 963-970.<br />
|-<br />
|[EL18c] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/1810.10950 Some examples of Picard groups of blocks]'', J. Algebra '''558''' (2020), 350-370.<br />
|-<br />
|[EL20] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/2006.11173 Donovan's conjecture and extensions by the centralizer of a defect group]'', J. Algebra '''582''' (2021), 157-176.<br />
|-<br />
|[EL23] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/2310.05734 Morita equivalence classes of <math>2</math>-blocks with abelian defect groups of rank <math>4</math>]'', [https://arxiv.org/abs/2310.05734 arxiv:2310.05734]<br />
|-<br />
|[Ei16] || '''F. Eisele''', ''Blocks with a generalized quaternion defect group and three simple modules over a <math>2</math>-adic ring'', J. Algebra '''456''' (2016), 294-322.<br />
|-<br />
|[Ei18] || '''F. Eisele''', ''[https://arxiv.org/abs/1807.05110 The Picard group of an order and Külshammer reduction]'', Algebr. Represent. Theory '''24''' (2021), 505-518.<br />
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|-<br />
|[EiLiv20] || '''F. Eisele and M. Livesey''', ''[https://arxiv.org/abs/2006.13837 Arbitrarily large Morita Frobenius numbers]'', Algebra Number Theory '''16''' (2022), 1889-1904.<br />
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|[Er82] || '''K. Erdmann''', ''Blocks whose defect groups are Klein four groups: a correction'', J. Algebra '''76''' (1982), 505-518.<br />
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|-<br />
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|-<br />
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|-<br />
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|- id="G"<br />
|[GMdelR21] || '''D. Garcia, l. Margolis and A. del Rio''', [https://arxiv.org/abs/2016.07231 ''Non-isomorphic 2-groups with isomorphic modular group algebras''], J. Reine Angew. Math. '''f783''' (2022), 269–274.<br />
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|-<br />
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|-<br />
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|-<br />
|[LiMa20] || '''M. Livesey and C. Marchi''', [https://arxiv.org/abs/2002.10571 ''On Picent for blocks with normal defect group''], [https://arxiv.org/abs/2002.10571 arXiv:2002.10571]<br />
|-<br />
|[LiMa20b] || '''M. Livesey and C. Marchi''', [https://arxiv.org/abs/2008.05857 ''Picard groups for blocks with normal defect groups and linear source bimodules''], [https://arxiv.org/abs/2008.05857 arXiv:2008.05857]<br />
|- id="M"<br />
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|-<br />
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|-<br />
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|-<br />
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|- id="N"<br />
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|- <br />
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|- id="O"<br />
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|- id="P"<br />
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|-<br />
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|-<br />
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|-<br />
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|-<br />
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|-<br />
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|- id="S"<br />
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|-<br />
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|-<br />
|[Sa12b] || '''B. Sambale''', ''Fusion systems on metacyclic 2-groups'', Osaka J. Math. '''49''' (2012), 325–329.<br />
|-<br />
|[Sa13] || '''B. Sambale''', ''Blocks with defect group <math>Q_{2^n} \times C_{2^m}</math> and <math>SD_{2^n} \times C_{2^m}</math>'', Algebr. Represent. Theory '''16''' (2013), 1717–1732.<br />
|-<br />
|[Sa13b] || '''B. Sambale''', ''Blocks with central product defect group <math>D_{2^n} ∗ C_{2^m}</math>'', Proc. Amer. Math. Soc. '''141''' (2013), 4057–4069.<br />
|-<br />
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|-<br />
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|-<br />
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|-<br />
|[Sa20] || '''B. Sambale''', [https://arxiv.org/abs/2005.13172 ''Blocks with small-dimensional basic algebra''], Bul. Aust. Math. Soc. '''103''' (2021), 461-474.<br />
|-<br />
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|-<br />
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|-<br />
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|-<br />
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|- id="T"<br />
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|- id="V"<br />
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|- id="W"<br />
|[Wa00]|| '''A. Watanabe''', ''A remark on a splitting theorem for blocks with abelian defect groups'', RIMS Kokyuroku '''1140''', Edited by H.Sasaki, Research Institute for Mathematical Sciences, Kyoto University (2000), 76-79.<br />
|-<br />
|[WZZ18] || '''Chao Wu, Kun Zhang and Yuanyang Zhou''', ''[https://arxiv.org/abs/1803.07262 Blocks with defect group <math>Z_{2^n} \times Z_{2^n} \times Z_{2^m}</math>]'', J. Algebra '''510''' (2018), 469-498.<br />
|}</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=References&diff=1222
References
2023-10-04T09:46:48Z
<p>Charles Eaton: </p>
<hr />
<div>[[#A|A,]] [[#B|B,]] [[#C|C,]] [[#D|D,]] [[#E|E,]] [[#F|F,]] [[#G|G,]] [[#H|H,]] [[#I|I,]] [[#J|J,]] [[#K|K,]] [[#L|L,]] [[#M|M,]] [[#N|N,]] [[#O|O,]] [[#P|P,]] [[#Q|Q,]] [[#R|R,]] [[#S|S,]] [[#T|T]] [[#U|U,]] [[#V|V,]] [[#W|W,]] [[#X|X,]] [[#Y|Y,]] [[#Z|Z,]] <br />
<br />
{| <br />
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|-<br />
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|-<br />
|[AE04] || '''Jianbei An and C. W. Eaton''', ''[https://arxiv.org/abs/2310.02150 "Morita equivalence classes of blocks with extraspecial defect groups <math>p_+^{1+2}</math>''], [https://arxiv.org/abs/2310.02150 arxiv.org/abs/2310.02150]<br />
|-<br />
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|-<br />
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|-<br />
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|-<br />
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|- id="B"<br />
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|-<br />
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|-<br />
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|-<br />
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|-<br />
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|-<br />
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|-<br />
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|-<br />
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|-<br />
|[CG12] || '''D. A. Craven and A. Glesser''', ''Fusion systems on small p-groups'', Trans. AMS '''364''' (2012) 5945-5967.<br />
|-<br />
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|-<br />
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|-<br />
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|- id="D"<br />
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|-<br />
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|-<br />
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|-<br />
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|- id="E"<br />
|[Ea16] || '''C. W. Eaton''', ''Morita equivalence classes of <math>2</math>-blocks of defect three'', Proc. AMS '''144''' (2016), 1961-1970.<br />
|-<br />
|[Ea18] || '''C. W. Eaton''', [https://arxiv.org/abs/1612.03485 ''Morita equivalence classes of blocks with elementary abelian defect groups of order 16''], [https://arxiv.org/abs/1612.03485 arXiv:1612.03485]<br />
|-<br />
|[EEL18] || '''C. W. Eaton, F. Eisele and M. Livesey''', [https://arxiv.org/abs/1809.08152 ''Donovan's conjecture, blocks with abelian defect groups and discrete valuation rings''], Math. Z. '''295''' (2020), 249-264.<br />
|-<br />
|[EKKS14] || '''C. W. Eaton, R. Kessar, B. Külshammer and B. Sambale''', ''<math>2</math>-blocks with abelian defect groups'', Adv. Math. '''254''' (2014), 706-735.<br />
|-<br />
|[EKS12] || '''C. W. Eaton, B. Külshammer and B. Sambale''', ''<math>2</math>-blocks with minimal nonabelian defect groups, II'', J. Group Theory '''15''' (2012), 311-321.<br />
|-<br />
|[EL18a] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/1709.04331 Classifying blocks with abelian defect groups of rank 3 for the prime 2]'', J. Algebra '''515''' (2018), 1-18.<br />
|-<br />
|[EL18b] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/1803.03539 Donovan's conjecture and blocks with abelian defect groups]'', Proc. AMS. '''147''' (2019), 963-970.<br />
|-<br />
|[EL18c] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/1810.10950 Some examples of Picard groups of blocks]'', J. Algebra '''558''' (2020), 350-370.<br />
|-<br />
|[EL20] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/2006.11173 Donovan's conjecture and extensions by the centralizer of a defect group]'', J. Algebra '''582''' (2021), 157-176.<br />
|-<br />
|[Ei16] || '''F. Eisele''', ''Blocks with a generalized quaternion defect group and three simple modules over a <math>2</math>-adic ring'', J. Algebra '''456''' (2016), 294-322.<br />
|-<br />
|[Ei18] || '''F. Eisele''', ''[https://arxiv.org/abs/1807.05110 The Picard group of an order and Külshammer reduction]'', Algebr. Represent. Theory '''24''' (2021), 505-518.<br />
|-<br />
|[Ei19] || '''F. Eisele''', ''[https://arxiv.org/abs/1908.00129 On the geometry of lattices and finiteness of Picard groups]'', [https://arxiv.org/abs/1908.00129 arXiv:1908.00129]<br />
|-<br />
|[EiLiv20] || '''F. Eisele and M. Livesey''', ''[https://arxiv.org/abs/2006.13837 Arbitrarily large Morita Frobenius numbers]'', [https://arxiv.org/abs/2006.13837 arXiv:2006.13837]<br />
|-<br />
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|-<br />
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|-<br />
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|-<br />
|[Er88b] || '''K. Erdmann''', ''Algebras and quaternion defect groups, II'', Math. Ann. '''281''' (1988), 561-582. <br />
|-<br />
|[Er88c] || '''K. Erdmann''', ''Algebras and semidihedral defect groups I'', Proc. LMS '''57''' (1988), 109-150. <br />
|-<br />
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|-<br />
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|- id="F"<br />
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|-<br />
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|- id="G"<br />
|[GMdelR21] || '''D. Garcia, l. Margolis and A. del Rio''', [https://arxiv.org/abs/2016.07231 ''Non-isomorphic 2-groups with isomorphic modular group algebras''], J. Reine Angew. Math. '''f783''' (2022), 269–274.<br />
|-<br />
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|-<br />
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|- id="H"<br />
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|-<br />
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|- <br />
|[HK00] || '''G. Hiss and R. Kessar''', ''Scopes reduction and Morita equivalence classes of blocks in finite classical groups'', J. Algebra '''230''' (2000), 378-423.<br />
|-<br />
|[HK05] || '''G. Hiss and R. Kessar''', ''Scopes reduction and Morita equivalence classes of blocks in finite classical groups II'', J. Algebra '''283''' (2005), 522-563.<br />
|-<br />
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|-<br />
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|- id="J"<br />
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|-<br />
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|- id="K"<br />
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|-<br />
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|-<br />
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|-<br />
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|-<br />
|[Ke05] || ''' R. Kessar''', ''A remark on Donovan's conjecture'', Arch. Math (Basel) '''82''' (2005), 391-394.<br />
|-<br />
|[KL18] || '''R. Kessar and M. Linckelmann''', [https://arxiv.org/abs/1705.07227 ''Descent of equivalences and character bijections''], [https://arxiv.org/abs/1705.07227 arXiv:1705.07227]<br />
|-<br />
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|-<br />
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|-<br />
|[KKW02] || '''S. Koshitani, N. Kunugi and K. Waki''', ''Broué's conjecture for non-principal 3-blocks of finite groups'', J. Pure and Applied Algebra '''173''' (2002), 177-211. <br />
|-<br />
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|-<br />
|[KoLa20] || '''S. Koshitani and C. Lassueur''', ''Splendid Morita equivalences for principal 2-blocks with dihedral defect groups'', Math. Z. '''294''' (2020), 639-666.<br />
|-<br />
|[KoLa20b] || '''S. Koshitani and C. Lassueur''', ''Splendid Morita equivalences for principal blocks with generalised quaternion defect groups'', J. Algebra '''558''' (2020), 523-533.<br />
|-<br />
|[KoLaSa22] || '''S. Koshitani, C. Lassueur and B. Sambale''', ''Splendid Morita equivalences for principal blocks with semidihedral defect groups'', Proceedings of the American Mathematical Society '''150''' (2022), 41-53.<br />
|-<br />
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|-<br />
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|-<br />
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|-<br />
|[Kü95] || '''B. Külshammer''', ''Donovan's conjecture, crossed products and algebraic group actions'', Israel J. Math. '''92''' (1995), 295-306.<br />
|-<br />
|[KS13] || '''B. Külshammer and B. Sambale''', ''The 2-blocks of defect 4'', Representation Theory '''17''' (2013), 226-236.<br />
|-<br />
|[Ku00] || '''N. Kunugi''', ''Morita equivalent 3-blocks of the 3-dimensional projective special linear groups'', Proc. LMS '''80''' (2000), 575-589.<br />
|-<br />
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|- id="L"<br />
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|-<br />
|[Li94] || '''M. Linckelmann''', ''The source algebras of blocks with a Klein four defect group'', J. Algebra '''167''' (1994), 821-854.<br />
|-<br />
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|-<br />
|[Li96] || '''M. Linckelmann''', ''The isomorphism problem for cyclic blocks and their source algebras'', Invent. Math. '''125''' (1996), 265-283.<br />
|-<br />
|[Li18] || '''M. Linckelmann''', [https://arxiv.org/abs/1805.08884 ''The strong Frobenius numbers for cyclic defect blocks are equal to one''], [https://arxiv.org/abs/1805.08884 arXiv:1805.08884]<br />
|-<br />
|[Li18b] || '''M. Linckelmann''', ''Finite-dimensional algebras arising as blocks of finite group algebras'', Contemporary Mathematics '''705''' (2018), 155-188.<br />
|-<br />
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|-<br />
|[Li18d] || '''M. Linckelmann''', ''The block theory of finite group algebras, Volume 2'', London Math. Soc. Student Texts '''92''', Cambridge University Press (2018).<br />
|-<br />
|[LM20] || '''M. Linckelmann and W. Murphy''', [https://arxiv.org/abs/2005.02223 ''A 9-dimensional algebra which is not a block of a finite group''], Quarterly Journal of Mathematics 72 (2021), 1077–1088<br />
|-<br />
|[Liv19] || '''M. Livesey''', [https://arxiv.org/abs/1907.12167 ''On Picard groups of blocks with normal defect groups''], J. Algebra '''566''' (2021), 94-118.<br />
|-<br />
|[LiMa20] || '''M. Livesey and C. Marchi''', [https://arxiv.org/abs/2002.10571 ''On Picent for blocks with normal defect group''], [https://arxiv.org/abs/2002.10571 arXiv:2002.10571]<br />
|-<br />
|[LiMa20b] || '''M. Livesey and C. Marchi''', [https://arxiv.org/abs/2008.05857 ''Picard groups for blocks with normal defect groups and linear source bimodules''], [https://arxiv.org/abs/2008.05857 arXiv:2008.05857]<br />
|- id="M"<br />
|[Mac] || '''N. Macgregor''', ''Morita equivalence classes of tame blocks of finite groups'', J. Algebra '''608''' (2022), 719-754.<br />
|-<br />
|[Mar] || '''C. Marchi''', ''Picard groups for blocks'', PhD thesis, University of Manchester (2022)<br />
|-<br />
|[Ma86] || '''U. Martin''', ''Almost all <math>p</math>-groups have automorphism group a <math>p</math>-group'', Bull. AMS '''15''' (1986), 78-82.<br />
|-<br />
|[McK19] || '''E. McKernon''', [https://arxiv.org/abs/1912.03222 ''2-Blocks whose defect group is homocyclic and whose inertial quotient contains a Singer cycle''], J. Algebra '''563''' (2020), 30–48.<br />
|-<br />
|[MS08] || '''J. Müller and M. Schaps''', ''The Broué conjecture for the faithful 3-blocks of <math>4.M_{22}</math>'', J. Algebra '''319''' (2008), 3588-3602.<br />
|- id="N"<br />
|[NS18] || '''G. Navarro and B. Sambale''', ''On the blockwise modular isomorphism problem'', Manuscripta Math. '''157''' (2018), 263-278.<br />
|- <br />
|[Ne02] || '''G. Nebe''', [http://www.math.rwth-aachen.de/~Gabriele.Nebe/papers/survey.pdf ''Group rings of finite groups over p-adic integers, some examples''], Proceedings of the conference Around Group rings (Edmonton) Resenhas '''5''' (2002), 329-350.<br />
|- id="O"<br />
|[Ok97] || '''T. Okuyama''', ''Some examples of derived equivalent blocks of finite groups'', preprint (1997).<br />
|- id="P"<br />
|[Pu88]|| '''L. Puig''', ''Nilpotent blocks and their source algebras'', Invent. Math. '''93''' (1988), 77-116.<br />
|-<br />
|[Pu94] || '''L. Puig''', ''On Joanna Scopes’ criterion of equivalence for blocks of symmetric groups'', Algebra Colloq. '''1''' (1994), 25-55.<br />
|-<br />
|[Pu99]|| '''L. Puig''', ''On the local structure of Morita and Rickard equivalences between Brauer blocks'', Progress in Math. '''178''', Birkhauser Verlag (1999).<br />
|-<br />
|[Pu09] || '''L. Puig''', ''Block source algebras in p-solvable groups'', Michigan Math. J. '''58''' (2009), 323-338.<br />
|- id="R"<br />
|[Ri96] || '''J. Rickard''', ''Splendid equivalences: derived categories and permutation modules'', Proc. London Math. Soc. '''72''' (1996), 331-358.<br />
|-<br />
|[Ro95] || '''R. Rouquier''', ''From stable equivalences to Rickard equivalences for blocks with cyclic defect'', Proceedings of Groups 1993, Galway-St. Andrews Conference, Vol. 2, London Math. Soc. Lecture Note Ser. '''212''', Cambridge University Press (1995), 512-523.<br />
|-<br />
|[Ru11] || '''P. Ruengrot''', ''Perfect isometry groups for blocks of finite groups'', PhD Thesis, University of Manchester (2011).<br />
|- id="S"<br />
|[Sa11] || '''B. Sambale''', ''<math>2</math>-blocks with minimal nonabelian defect groups'', J. Algebra '''337''' (2011), 261–284.<br />
|-<br />
|[Sa12] || '''B. Sambale''', ''Blocks with defect group <math>D_{2^n} \times C_{2^m}</math>'', J. Pure Appl. Algebra '''216''' (2012), 119–125.<br />
|-<br />
|[Sa12b] || '''B. Sambale''', ''Fusion systems on metacyclic 2-groups'', Osaka J. Math. '''49''' (2012), 325–329.<br />
|-<br />
|[Sa13] || '''B. Sambale''', ''Blocks with defect group <math>Q_{2^n} \times C_{2^m}</math> and <math>SD_{2^n} \times C_{2^m}</math>'', Algebr. Represent. Theory '''16''' (2013), 1717–1732.<br />
|-<br />
|[Sa13b] || '''B. Sambale''', ''Blocks with central product defect group <math>D_{2^n} ∗ C_{2^m}</math>'', Proc. Amer. Math. Soc. '''141''' (2013), 4057–4069.<br />
|-<br />
|[Sa13c] || '''B. Sambale''', ''Further evidence for conjectures in block theory'', Algebra Number Theory '''7''' (2013), 2241–2273. <br />
|-<br />
|[Sa14] || '''B. Sambale''', ''Blocks of Finite Groups and Their Invariants'', Lecture Notes in Mathematics, Springer (2014).<br />
|-<br />
|[Sa16] || '''B. Sambale''', ''2-blocks with minimal nonabelian defect groups III'', Pacific J. Math. '''280''' (2016), 475–487.<br />
|-<br />
|[Sa20] || '''B. Sambale''', [https://arxiv.org/abs/2005.13172 ''Blocks with small-dimensional basic algebra''], Bul. Aust. Math. Soc. '''103''' (2021), 461-474.<br />
|-<br />
|[SSS98] || '''M. Schaps, D. Shapira and O. Shlomo''', ''Quivers of blocks with normal defect groups'', Proc. Symp. in Pure Mathematics '''63''', Amer. Math. Soc. (1998), 497-510.<br />
|-<br />
|[Sc91] || '''J. Scopes''', ''Cartan matrices and Morita equivalence for blocks of the symmetric groups'', J. Algebra '''142''' (1991), 441-455.<br />
|-<br />
|[Sh20] || '''V. Shalotenko''', ''Bounds on the dimension of Ext for finite groups of Lie type'', J. Algebra '''550''' (2020), 266-289.<br />
|-<br />
|[St02] || '''R. Stancu''', ''Almost all generalized extraspecial p-groups are resistant'', J. Algebra '''249''' (2002), 120-126.<br />
|-<br />
|[St06] || '''R. Stancu''', ''Control of fusion in fusion systems'', J. Algebra and its Applications '''5''' (2006), 817-837. <br />
|- id="T"<br />
|[Th93] || '''J. Thévenaz''', ''Most finite groups are <math>p</math>-nilpotent'', Exposition. Math. '''11''' (1993), 359-363.<br />
|- id="V"<br />
|[vdW91] || '''R. van der Waall''', ''On p-nilpotent forcing groups'', Indag. Mathem., N.S., '''2''' (1991), 367-384.<br />
|- id="W"<br />
|[Wa00]|| '''A. Watanabe''', ''A remark on a splitting theorem for blocks with abelian defect groups'', RIMS Kokyuroku '''1140''', Edited by H.Sasaki, Research Institute for Mathematical Sciences, Kyoto University (2000), 76-79.<br />
|-<br />
|[WZZ18] || '''Chao Wu, Kun Zhang and Yuanyang Zhou''', ''[https://arxiv.org/abs/1803.07262 Blocks with defect group <math>Z_{2^n} \times Z_{2^n} \times Z_{2^m}</math>]'', J. Algebra '''510''' (2018), 469-498.<br />
|}</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=5_%2B%5E3&diff=1221
5 +^3
2023-10-04T09:36:21Z
<p>Charles Eaton: /* Blocks with defect group 5_+^{1+2} */</p>
<hr />
<div>== Blocks with defect group <math>5_+^{1+2}</math> ==<br />
<br />
The Morita equivalence classes of blocks with this defect group are classified in [[References#A|[AE23]]]. Blocks in the same Morita equivalence class have the same fusion, and so the same inertial quotient, and the same Külshammer-Puig class.<br />
<br />
{| class="wikitable"<br />
|-<br />
! scope="col"| Class<br />
! scope="col"| Representative<br />
! scope="col"| [[Glossary|# lifts / <math>\mathcal{O}</math>]]<br />
! scope="col"| <math>k_0(B)</math><br />
! scope="col"| <math>k_1(B)</math><br />
! scope="col"| <math>l(B)</math><br />
! scope="col"| Inertial quotient<br />
! scope="col"| <math>{\rm Pic}_\mathcal{O}(B)</math><br />
! scope="col"| <math>{\rm Pic}_k(B)</math><br />
! scope="col"| Notes<br />
<br />
|-<br />
|[[M(125,3,1)]] || <math>k5_+^{1+2}</math> || 1 || 20 || 4 ||1 || <math>1</math> || || || <br />
|-<br />
|[[M(125,3,2)]] || <math>k5_+^{1+2}:C_2</math> || 1 || 20 || 2 ||2 || <math>C_2</math> || || || SmallGroup(250,5)<br />
|-<br />
|[[M(125,3,3)]] || <math>k5_+^{1+2}:C_2</math> || 1 || 14 || 8 ||2 || <math>C_2</math> || || || SmallGroup(250,8)<br />
|-<br />
|[[M(125,3,4)]] || <math>k5_+^{1+2}:C_3</math> || 1 || 11 || 12 ||3 || <math>C_3</math> || || || SmallGroup(375,2)<br />
|-<br />
|[[M(125,3,5)]] || <math>k5_+^{1+2}:C_4</math> || 1 || 25 || 1 ||4 || <math>C_4</math> || || || SmallGroup(500,17)<br />
|-<br />
|[[M(125,3,6)]] || <math>k5_+^{1+2}:C_4</math> || 1 || 13 || 1 ||4 || <math>C_4</math> || || || SmallGroup(500,21)<br />
|-<br />
|[[M(125,3,7)]] || <math>k(C_5 \times C_5):SL_2(5)</math> || 1 || 13 || 1 ||5 || <math>C_4</math> || || || Inertial quotient as in M(125,3,6)<br />
|-<br />
|[[M(125,3,8)]] || <math>k5_+^{1+2}:C_4</math> || 1 || 10 || 4 ||4 || <math>C_4</math> || || || SmallGroup(500,23)<br />
|-<br />
|[[M(125,3,9)]] || <math>k5_+^{1+2}:C_4</math> || 1 || 10 || 16 ||4 || <math>C_4</math> || || || SmallGroup(500,25)<br />
|-<br />
|[[M(125,3,10)]] || <math>k5_+^{1+2}:(C_2 \times C_2)</math> || 1 || 16 || 4 ||4 || <math>C_2 \times C_2</math> || || || SmallGroup(500,27)<br />
|-<br />
|[[M(125,3,11)]] || Faithful block of <math>k5_+^{1+2}:Q_8</math>, in which <math>Z(Q_8)</math> acts trivially || 1 || 13 || 4 ||1 || <math>C_2 \times C_2</math> || || || SmallGroup(1000,42)<br />
|-<br />
|[[M(125,3,12)]] || <math>k5_+^{1+2}:C_6</math> || 1 || 10 || 24 ||6 || <math>C_6</math> || || || SmallGroup(750,6)<br />
|-<br />
|[[M(125,3,13)]] || <math>k5_+^{1+2}:S_3</math> || 1 || 13 || 6 ||3 || <math>S_3</math> || || || SmallGroup(750,5)<br />
|-<br />
|[[M(125,3,14)]] || <math>k5_+^{1+2}:C_8</math> || 1 || 11 || 2 ||8 || <math>C_8</math> || || || SmallGroup(1000,86)<br />
|-<br />
|[[M(125,4,15)]] || <math>B_0(kSU_3(5))</math> || 1 || 11 || 2 || 8 || <math>C_8</math> || || ||<br />
|-<br />
|[[M(125,3,16)]] || Faithful block of <math>k(3.SU_3(5))</math> || 1 || 11 || 2 || 8 || <math>C_8</math> || || ||<br />
|-<br />
|[[M(125,3,17)]] || <math>k5_+^{1+2}:(C_4 \times C_2)</math> || 1 || 20 || 2 ||8 || <math>C_4 \times C_2</math> || || || SmallGroup(1000,89)<br />
|-<br />
|[[M(125,3,18)]] || Faithful block of <math>k5_+^{1+2}:M_4(2)</math>, in which <math>M_4(2)'</math> acts trivially || 1 || 14 || 2 ||2 || <math>C_2 \times C_2</math> || || || SmallGroup(2000,250)<br />
|-<br />
|-[[M(125,3,19)]] || <math>k((C_5 \times C_5):SL_2(5).2)</math> || 1 || 20 || 2 || 10 || <math>C_4 \times C_2</math> || Inertial quotient as in M(125,3,17)<br />
|-<br />
|[[M(125,3,20)]] || <math>k5_+^{1+2}:(C_4 \times C_2)</math> || 1 || 14 || 8 ||8 || <math>C_4 \times C_2</math> || || || SmallGroup(1000,91)<br />
|-<br />
|[[M(125,3,21)]] || Faithful block of <math>k5_+^{1+2}:M_4(2)</math>, in which <math>M_4(2)'</math> acts trivially || 1 || 8 || 8 ||2 || <math>C_2 \times C_2</math> || || || SmallGroup(2000,264)<br />
|-<br />
|[[M(125,3,22)]] || <math>k5_+^{1+2}:D_8</math> || 1 || 14 || 8 ||5 || <math>D_8</math> || || || SmallGroup(1000,92)<br />
|-<br />
|[[M(125,3,23)]] || Faithful block of <math>k5_+^{1+2}:D_{16}</math>, in which <math>Z(D_{16})</math> acts trivially || 1 || 11 || 8 ||2 || <math>D_8</math> || || || <br />
|-<br />
|[[M(125,3,24)]] || <math>k5_+^{1+2}:Q_8</math> || 1 || 8 || 20 ||5 || <math>Q_8</math> || || || SmallGroup(1000,93)<br />
|-<br />
|[[M(125,3,25)]] || <math>k5_+^{1+2}:(C_3:C_4)</math> || 1 || 8 || 24 ||6 || <math>C_3:C_4</math> || || || SmallGroup(1500,35)<br />
|-<br />
|[[M(125,3,26)]] || <math>k5_+^{1+2}:C_{12}</math> || 1 || 14 || 12 ||12 || <math>C_{12}</math> || || || SmallGroup(1500,36)<br />
|-<br />
|[[M(125,3,27)]] || <math>k5_+^{1+2}:D_{12}</math> || 1 || 14 || 12 ||6 || <math>D_{12}</math> || || || SmallGroup(1000,37)<br />
|-<br />
|[[M(125,3,28)]] || Faithful block of <math>k5_+^{1+2}:D_{24}</math>, in which <math>Z(D_{24})</math> acts trivially || 1 || 10 || 4 ||1 || <math>D_{12}</math> || || || <br />
|-<br />
|[[M(125,3,29)]] || <math>k5_+^{1+2}:(C_4 \times C_4)</math> || 1 || 25 || 4 ||16 || <math>C_4 \times C_4</math> || || || SmallGroup(2000,473)<br />
|-<br />
|[[M(125,3,30)]] || Faithful block of <math>k5_+^{1+2}:(C_2.(C_4 \times C_4))</math>, in which <math>(C_2.(C_4 \times C_4))'</math> acts trivially || 1 || 13 || 4 ||4 || <math>C_4 \times C_4</math> || || || <math>C_2.(C_4 \times C_4)</math> is SmallGroup(32,2)<br />
|-<br />
|[[M(125,3,31)]] || Faithful block of <math>k5_+^{1+2}:((C_4 \times C_4):C_4)</math>, in which <math>Z((C_4 \times C_4):C_4)</math> acts trivially || 1 || 10 || 4 ||1 || <math>C_4 \times C_4</math> || || || <math>(C_4 \times C_4):C_4</math> is SmallGroup(64,18)<br />
|-<br />
|[[M(125,3,32)]] || <math>k(C_5 \times C_5):GL_2(5)</math> || 1 || 25 || 4 ||20 || <math>C_4 \times C_4</math> || || || <br />
|-<br />
|[[M(125,3,33)]] || <math>B_0(kSL_3(5))</math> || 1 || 25 || 4 ||24 || <math>C_4 \times C_4</math> || || || <br />
|-<br />
|[[M(125,3,34)]] || <math>k5_+^{1+2}:M_4(2)</math> || 1 || 13 || 4 ||10 || <math>M_4(2)</math> || || || SmallGroup(2000,474)<br />
|-<br />
|[[M(125,3,35)]] || <math>B_0(kSU_3(5).2)</math> || 1 || 13 || 4 ||10 || <math>M_4(2)</math> || || || <br />
|-<br />
|[[M(125,3,36)]] || <math>B_0(HS)</math> || 1 || 13 || 4 ||10 || <math>M_4(2)</math> || || || <br />
|-<br />
|[[M(125,3,37)]] || Faithful block of <math>k(2.HS)</math> || 1 || 13 || 4 ||10 || <math>M_4(2)</math> || || || <br />
|-<br />
|[[M(125,3,38)]] || <math>k5_+^{1+2}:(C_4 \circ D_8)</math> || 1 || 16 || 10 ||10 || <math>C_4 \circ D_8</math> || || ||<br />
|-<br />
|[[M(125,3,39)]] || Faithful block of <math>k5_+^{1+2}:H</math>, in which <math>C_2 \times C_2 \cong Z \leq Z(H)</math> acts trivially (with <math>H \cong (C_2 \times C_2 \times C_2):D_8</math> and <math>H/Z \cong C_4 \circ D_8</math>) || 1 || 10 || 10 ||4 || <math>C_4 \circ D_8</math> || || || <math>H</math> is SmallGroup(64,73)<br />
|-<br />
|[[M(125,3,40)]] || <math>k5_+^{1+2}:(C_3:C_8)</math> || 1 || 13 || 6 ||12 || <math>C_3:C_8</math> || || ||<br />
|-<br />
|[[M(125,3,41)]] || <math>B_0(kMcL)</math> || 1 || 13 || 6 ||12 || <math>C_3:C_8</math> || || ||<br />
|-<br />
|[[M(125,3,42)]] || <math>k(3.McL)</math> || 1 || 13 || 6 ||12 || <math>C_3:C_8</math> || || ||<br />
|-<br />
|[[M(125,3,43)]] || <math>k5_+^{1+2}:C_{24}</math> || 1 || 25 || 6 ||24 || <math>C_{24}</math> || || ||<br />
|-<br />
|[[M(125,3,44)]] || <math>B_0(SU_3(5).3)</math> || 1 || 25 || 6 ||24 || <math>C_{24}</math> || || ||<br />
|-<br />
|[[M(125,3,45)]] || Faithful block of <math>kSU_3(5).3</math> || 1 || 25 || 6 ||24 || <math>C_{24}</math> || || ||<br />
|-<br />
|[[M(125,3,46)]] || <math>k5_+^{1+2}:SL_2(3)</math> || 1 || 8 || 28 ||7 || <math>C_{24}</math> || || ||<br />
|-<br />
|[[M(125,3,47)]] || <math>k5_+^{1+2}:(C_4 \times S_3)</math> || 1 || 16 || 12 ||12 || <math>C_4 \times S_3</math> || || ||<br />
|-<br />
|[[M(125,3,48)]] || Faithful block <math>k5_+^{1+2}:(C_8:S_3)</math>, in which <math>C_2 \cong Z \leq Z(C_8:S_3)</math> acts trivially (with <math>(C_8:S_3)/Z \cong C_4 \times S_3</math>) || 1 || 10 || 12 ||6 || <math>C_4 \times S_3</math> || || ||<br />
|-<br />
|[[M(125,3,49)]] || <math>k5_+^{1+2}:(C_4 \wr C_2)</math> || 1 || 20 || 5 ||14 || <math>C_4 \wr C_2</math> || || ||<br />
|-<br />
|[[M(125,3,50)]] || Faithful block of <math>k5_+^{1+2}:H</math>, in which <math>C_2 \cong Z \leq Z(H)</math> acts trivially (with <math>H=</math>SmallGroup(64,10) and <math>H/Z \cong C_4 \wr C_2</math>) || 1 || 11 || 5 ||5 || <math>C_4 \wr C_2</math> || || ||<br />
|-<br />
|[[M(125,3,51)]] || <math>B_0(kSL_3(5).2)</math> || 1 || 20 || 5 ||18 || <math>C_4 \wr C_2</math> || || ||<br />
|-<br />
|[[M(125,3,52)]] || <math>B_0(kHS.2)</math> || 1 || 20 || 5 ||18 || <math>C_4 \wr C_2</math> || || ||<br />
|-<br />
|[[M(125,3,53)]] || Faithful block of <math>k2.HS.2</math> || 1 || 20 || 5 ||18 || <math>C_4 \wr C_2</math> || || ||<br />
|-<br />
|[[M(125,3,54)]] || <math>B_0(kRu)</math> || 1 || 20 || 5 ||18 || <math>C_4 \wr C_2</math> || || ||<br />
|-<br />
|[[M(125,3,55)]] || Faithful block of <math>k2.Ru</math> || 1 || 20 || 5 ||18 || <math>C_4 \wr C_2</math> || || ||<br />
|-<br />
|[[M(125,3,56)]] || <math>k5_+^{1+2}:(C_8:S_3)</math> || 1 || 20 || 6 ||18 || <math>C_8:S_3</math> || || ||<br />
|-<br />
|[[M(125,3,57)]] || <math>B_0(kMcL.2)</math> || 1 || 20 || 6 ||18 || <math>C_8:S_3</math> || || ||<br />
|-<br />
|[[M(125,3,58)]] || <math>B_0(kCo_3)</math> || 1 || 20 || 6 ||18 || <math>C_8:S_3</math> || || ||<br />
|-<br />
|[[M(125,3,59)]] || <math>k5_+^{1+2}:(C_4 \circ SL_2(3))</math> || 1 || 16 || 14 ||14 || <math>C_4 \circ SL_2(3)</math> || || ||<br />
|-<br />
|[[M(125,3,60)]] || <math>k5_+^{1+2}:U_2(3)</math> || 1 || 20 || 7 ||16 || <math>U_2(3)</math> || || ||<br />
|-<br />
|[[M(125,3,61)]] || <math>B_0(kCo_2)</math> || 1 || 20 || 7 ||16 || <math>U_2(3)</math> || || ||<br />
|-<br />
|[[M(125,3,62)]] || <math>B_0(kTh)</math> || 1 || 20 || 7 ||20 || <math>U_2(3)</math> || || ||<br />
|}</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=5_%2B%5E3&diff=1220
5 +^3
2023-10-03T15:47:45Z
<p>Charles Eaton: </p>
<hr />
<div>== Blocks with defect group <math>5_+^{1+2}</math> ==<br />
<br />
[[Image:under-construction.png|50px|left]]<br />
<br />
The Morita equivalence classes of blocks with this defect group are classified in [[References#A|[AE23]]]. Blocks in the same Morita equivalence class have the same fusion, and so the same inertial quotient, and the same Külshammer-Puig class.<br />
<br />
{| class="wikitable"<br />
|-<br />
! scope="col"| Class<br />
! scope="col"| Representative<br />
! scope="col"| [[Glossary|# lifts / <math>\mathcal{O}</math>]]<br />
! scope="col"| <math>k_0(B)</math><br />
! scope="col"| <math>k_1(B)</math><br />
! scope="col"| <math>l(B)</math><br />
! scope="col"| Inertial quotient<br />
! scope="col"| <math>{\rm Pic}_\mathcal{O}(B)</math><br />
! scope="col"| <math>{\rm Pic}_k(B)</math><br />
! scope="col"| Notes<br />
<br />
|-<br />
|[[M(125,3,1)]] || <math>k5_+^{1+2}</math> || 1 || 20 || 4 ||1 || <math>1</math> || || || <br />
|-<br />
|[[M(125,3,2)]] || <math>k5_+^{1+2}:C_2</math> || 1 || 20 || 2 ||2 || <math>C_2</math> || || || SmallGroup(250,5)<br />
|-<br />
|[[M(125,3,3)]] || <math>k5_+^{1+2}:C_2</math> || 1 || 14 || 8 ||2 || <math>C_2</math> || || || SmallGroup(250,8)<br />
|-<br />
|[[M(125,3,4)]] || <math>k5_+^{1+2}:C_3</math> || 1 || 11 || 12 ||3 || <math>C_3</math> || || || SmallGroup(375,2)<br />
|-<br />
|[[M(125,3,5)]] || <math>k5_+^{1+2}:C_4</math> || 1 || 25 || 1 ||4 || <math>C_4</math> || || || SmallGroup(500,17)<br />
|-<br />
|[[M(125,3,6)]] || <math>k5_+^{1+2}:C_4</math> || 1 || 13 || 1 ||4 || <math>C_4</math> || || || SmallGroup(500,21)<br />
|-<br />
|[[M(125,3,7)]] || <math>k(C_5 \times C_5):SL_2(5)</math> || 1 || 13 || 1 ||5 || <math>C_4</math> || || || Inertial quotient as in M(125,3,6)<br />
|-<br />
|[[M(125,3,8)]] || <math>k5_+^{1+2}:C_4</math> || 1 || 10 || 4 ||4 || <math>C_4</math> || || || SmallGroup(500,23)<br />
|-<br />
|[[M(125,3,9)]] || <math>k5_+^{1+2}:C_4</math> || 1 || 10 || 16 ||4 || <math>C_4</math> || || || SmallGroup(500,25)<br />
|-<br />
|[[M(125,3,10)]] || <math>k5_+^{1+2}:(C_2 \times C_2)</math> || 1 || 16 || 4 ||4 || <math>C_2 \times C_2</math> || || || SmallGroup(500,27)<br />
|-<br />
|[[M(125,3,11)]] || Faithful block of <math>k5_+^{1+2}:Q_8</math>, in which <math>Z(Q_8)</math> acts trivially || 1 || 13 || 4 ||1 || <math>C_2 \times C_2</math> || || || SmallGroup(1000,42)<br />
|-<br />
|[[M(125,3,12)]] || <math>k5_+^{1+2}:C_6</math> || 1 || 10 || 24 ||6 || <math>C_6</math> || || || SmallGroup(750,6)<br />
|-<br />
|[[M(125,3,13)]] || <math>k5_+^{1+2}:S_3</math> || 1 || 13 || 6 ||3 || <math>S_3</math> || || || SmallGroup(750,5)<br />
|-<br />
|[[M(125,3,14)]] || <math>k5_+^{1+2}:C_8</math> || 1 || 11 || 2 ||8 || <math>C_8</math> || || || SmallGroup(1000,86)<br />
|-<br />
|[[M(125,4,15)]] || <math>B_0(kSU_3(5))</math> || 1 || 11 || 2 || 8 || <math>C_8</math> || || ||<br />
|-<br />
|[[M(125,3,16)]] || Faithful block of <math>k(3.SU_3(5))</math> || 1 || 11 || 2 || 8 || <math>C_8</math> || || ||<br />
|-<br />
|[[M(125,3,17)]] || <math>k5_+^{1+2}:(C_4 \times C_2)</math> || 1 || 20 || 2 ||8 || <math>C_4 \times C_2</math> || || || SmallGroup(1000,89)<br />
|-<br />
|[[M(125,3,18)]] || Faithful block of <math>k5_+^{1+2}:M_4(2)</math>, in which <math>M_4(2)'</math> acts trivially || 1 || 14 || 2 ||2 || <math>C_2 \times C_2</math> || || || SmallGroup(2000,250)<br />
|-<br />
|-[[M(125,3,19)]] || <math>k((C_5 \times C_5):SL_2(5).2)</math> || 1 || 20 || 2 || 10 || <math>C_4 \times C_2</math> || Inertial quotient as in M(125,3,17)<br />
|-<br />
|[[M(125,3,20)]] || <math>k5_+^{1+2}:(C_4 \times C_2)</math> || 1 || 14 || 8 ||8 || <math>C_4 \times C_2</math> || || || SmallGroup(1000,91)<br />
|-<br />
|[[M(125,3,21)]] || Faithful block of <math>k5_+^{1+2}:M_4(2)</math>, in which <math>M_4(2)'</math> acts trivially || 1 || 8 || 8 ||2 || <math>C_2 \times C_2</math> || || || SmallGroup(2000,264)<br />
|-<br />
|[[M(125,3,22)]] || <math>k5_+^{1+2}:D_8</math> || 1 || 14 || 8 ||5 || <math>D_8</math> || || || SmallGroup(1000,92)<br />
|-<br />
|[[M(125,3,23)]] || Faithful block of <math>k5_+^{1+2}:D_{16}</math>, in which <math>Z(D_{16})</math> acts trivially || 1 || 11 || 8 ||2 || <math>D_8</math> || || || <br />
|-<br />
|[[M(125,3,24)]] || <math>k5_+^{1+2}:Q_8</math> || 1 || 8 || 20 ||5 || <math>Q_8</math> || || || SmallGroup(1000,93)<br />
|-<br />
|[[M(125,3,25)]] || <math>k5_+^{1+2}:(C_3:C_4)</math> || 1 || 8 || 24 ||6 || <math>C_3:C_4</math> || || || SmallGroup(1500,35)<br />
|-<br />
|[[M(125,3,26)]] || <math>k5_+^{1+2}:C_{12}</math> || 1 || 14 || 12 ||12 || <math>C_{12}</math> || || || SmallGroup(1500,36)<br />
|-<br />
|[[M(125,3,27)]] || <math>k5_+^{1+2}:D_{12}</math> || 1 || 14 || 12 ||6 || <math>D_{12}</math> || || || SmallGroup(1000,37)<br />
|-<br />
|[[M(125,3,28)]] || Faithful block of <math>k5_+^{1+2}:D_{24}</math>, in which <math>Z(D_{24})</math> acts trivially || 1 || 10 || 4 ||1 || <math>D_{12}</math> || || || <br />
|-<br />
|[[M(125,3,29)]] || <math>k5_+^{1+2}:(C_4 \times C_4)</math> || 1 || 25 || 4 ||16 || <math>C_4 \times C_4</math> || || || SmallGroup(2000,473)<br />
|-<br />
|[[M(125,3,30)]] || Faithful block of <math>k5_+^{1+2}:(C_2.(C_4 \times C_4))</math>, in which <math>(C_2.(C_4 \times C_4))'</math> acts trivially || 1 || 13 || 4 ||4 || <math>C_4 \times C_4</math> || || || <math>C_2.(C_4 \times C_4)</math> is SmallGroup(32,2)<br />
|-<br />
|[[M(125,3,31)]] || Faithful block of <math>k5_+^{1+2}:((C_4 \times C_4):C_4)</math>, in which <math>Z((C_4 \times C_4):C_4)</math> acts trivially || 1 || 10 || 4 ||1 || <math>C_4 \times C_4</math> || || || <math>(C_4 \times C_4):C_4</math> is SmallGroup(64,18)<br />
|-<br />
|[[M(125,3,32)]] || <math>k(C_5 \times C_5):GL_2(5)</math> || 1 || 25 || 4 ||20 || <math>C_4 \times C_4</math> || || || <br />
|-<br />
|[[M(125,3,33)]] || <math>B_0(kSL_3(5))</math> || 1 || 25 || 4 ||24 || <math>C_4 \times C_4</math> || || || <br />
|-<br />
|[[M(125,3,34)]] || <math>k5_+^{1+2}:M_4(2)</math> || 1 || 13 || 4 ||10 || <math>M_4(2)</math> || || || SmallGroup(2000,474)<br />
|-<br />
|[[M(125,3,35)]] || <math>B_0(kSU_3(5).2)</math> || 1 || 13 || 4 ||10 || <math>M_4(2)</math> || || || <br />
|-<br />
|[[M(125,3,36)]] || <math>B_0(HS)</math> || 1 || 13 || 4 ||10 || <math>M_4(2)</math> || || || <br />
|-<br />
|[[M(125,3,37)]] || Faithful block of <math>k(2.HS)</math> || 1 || 13 || 4 ||10 || <math>M_4(2)</math> || || || <br />
|-<br />
|[[M(125,3,38)]] || <math>k5_+^{1+2}:(C_4 \circ D_8)</math> || 1 || 16 || 10 ||10 || <math>C_4 \circ D_8</math> || || ||<br />
|-<br />
|[[M(125,3,39)]] || Faithful block of <math>k5_+^{1+2}:H</math>, in which <math>C_2 \times C_2 \cong Z \leq Z(H)</math> acts trivially (with <math>H \cong (C_2 \times C_2 \times C_2):D_8</math> and <math>H/Z \cong C_4 \circ D_8</math>) || 1 || 10 || 10 ||4 || <math>C_4 \circ D_8</math> || || || <math>H</math> is SmallGroup(64,73)<br />
|-<br />
|[[M(125,3,40)]] || <math>k5_+^{1+2}:(C_3:C_8)</math> || 1 || 13 || 6 ||12 || <math>C_3:C_8</math> || || ||<br />
|-<br />
|[[M(125,3,41)]] || <math>B_0(kMcL)</math> || 1 || 13 || 6 ||12 || <math>C_3:C_8</math> || || ||<br />
|-<br />
|[[M(125,3,42)]] || <math>k(3.McL)</math> || 1 || 13 || 6 ||12 || <math>C_3:C_8</math> || || ||<br />
|-<br />
|[[M(125,3,43)]] || <math>k5_+^{1+2}:C_{24}</math> || 1 || 25 || 6 ||24 || <math>C_{24}</math> || || ||<br />
|-<br />
|[[M(125,3,44)]] || <math>B_0(SU_3(5).3)</math> || 1 || 25 || 6 ||24 || <math>C_{24}</math> || || ||<br />
|-<br />
|[[M(125,3,45)]] || Faithful block of <math>kSU_3(5).3</math> || 1 || 25 || 6 ||24 || <math>C_{24}</math> || || ||<br />
|-<br />
|[[M(125,3,46)]] || <math>k5_+^{1+2}:SL_2(3)</math> || 1 || 8 || 28 ||7 || <math>C_{24}</math> || || ||<br />
|}</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=5_%2B%5E3&diff=1219
5 +^3
2023-10-03T14:00:03Z
<p>Charles Eaton: Created page with "== Blocks with defect group <math>5_+^{1+2}</math> == left The Morita equivalence classes of blocks with this defect group are classifi..."</p>
<hr />
<div>== Blocks with defect group <math>5_+^{1+2}</math> ==<br />
<br />
[[Image:under-construction.png|50px|left]]<br />
<br />
The Morita equivalence classes of blocks with this defect group are classified in [[References#A|[AE23]]]. Blocks in the same Morita equivalence class have the same fusion, and so the same inertial quotient, and the same Külshammer-Puig class.<br />
<br />
{| class="wikitable"<br />
|-<br />
! scope="col"| Class<br />
! scope="col"| Representative<br />
! scope="col"| [[Glossary|# lifts / <math>\mathcal{O}</math>]]<br />
! scope="col"| <math>k_0(B)</math><br />
! scope="col"| <math>k_1(B)</math><br />
! scope="col"| <math>l(B)</math><br />
! scope="col"| Inertial quotient<br />
! scope="col"| <math>{\rm Pic}_\mathcal{O}(B)</math><br />
! scope="col"| <math>{\rm Pic}_k(B)</math><br />
! scope="col"| Notes<br />
<br />
|-<br />
|[[M(125,3,1)]] || <math>k5_+^{1+2}</math> || 1 || 20 || 4 ||1 || <math>1</math> || || || <br />
|-<br />
|[[M(125,3,2)]] || <math>k5_+^{1+2}:C_2</math> || 1 || 20 || 2 ||2 || <math>C_2</math> || || || SmallGroup(250,5)<br />
|-<br />
|[[M(125,3,3)]] || <math>k5_+^{1+2}:C_2</math> || 1 || 14 || 8 ||2 || <math>C_2</math> || || || SmallGroup(250,8)<br />
|-<br />
|[[M(125,3,4)]] || <math>k5_+^{1+2}:C_3</math> || 1 || 11 || 12 ||3 || <math>C_3</math> || || || SmallGroup(375,2)<br />
|-<br />
|[[M(125,3,5)]] || <math>k5_+^{1+2}:C_4</math> || 1 || 25 || 1 ||4 || <math>C_4</math> || || || SmallGroup(500,17)<br />
|-<br />
|[[M(125,3,6)]] || <math>k5_+^{1+2}:C_4</math> || 1 || 13 || 1 ||4 || <math>C_4</math> || || || SmallGroup(500,21)<br />
|-<br />
|[[M(125,3,7)]] || <math>k(C_5 \times C_5):SL_2(5)</math> || 1 || 13 || 1 ||5 || <math>C_4</math> || || || Inertial quotient as in M(125,3,6)<br />
|-<br />
|[[M(125,3,8)]] || <math>k5_+^{1+2}:C_4</math> || 1 || 10 || 4 ||4 || <math>C_4</math> || || || SmallGroup(500,23)<br />
|-<br />
|[[M(125,3,9)]] || <math>k5_+^{1+2}:C_4</math> || 1 || 10 || 16 ||4 || <math>C_4</math> || || || SmallGroup(500,25)<br />
|-<br />
|[[M(125,3,10)]] || <math>k5_+^{1+2}:(C_2 \times C_2)</math> || 1 || 16 || 4 ||4 || <math>C_2 \times C_2</math> || || || SmallGroup(500,27)<br />
|-<br />
|[[M(125,3,11)]] || Faithful block of <math>k5_+^{1+2}:Q_8</math>, in which <math>Z(Q_8)</math> acts trivially || 1 || 13 || 4 ||1 || <math>C_2 \times C_2</math> || || || SmallGroup(1000,42)<br />
|-<br />
|[[M(125,3,12)]] || <math>k5_+^{1+2}:C_6</math> || 1 || 10 || 24 ||6 || <math>C_6</math> || || || SmallGroup(750,6)<br />
|-<br />
|[[M(125,3,13)]] || <math>k5_+^{1+2}:S_3</math> || 1 || 13 || 6 ||3 || <math>S_3</math> || || || SmallGroup(750,5)<br />
|-<br />
|[[M(125,3,14)]] || <math>k5_+^{1+2}:C_8</math> || 1 || 11 || 2 ||8 || <math>C_8</math> || || || SmallGroup(1000,86)<br />
|-<br />
|[[M(125,4,15)]] || <math>B_0(kSU_3(5))</math> || 1 || 11 || 2 || 8 || <math>C_8</math> || || ||<br />
|-<br />
|[[M(125,3,16)]] || Faithful block of maximal defect of <math>k(3.SU_3(5))</math> || 1 || 11 || 2 || 8 || <math>C_8</math> || || ||<br />
|-<br />
|[[M(125,3,17)]] || <math>k5_+^{1+2}:(C_4 \times C_2)</math> || 1 || 20 || 2 ||8 || <math>C_4 \times C_2</math> || || || SmallGroup(1000,89)<br />
|-<br />
|[[M(125,3,18)]] || Faithful block of <math>k5_+^{1+2}:M_4(2)</math>, in which <math>M_4(2)'</math> acts trivially || 1 || 14 || 2 ||2 || <math>C_2 \times C_2</math> || || || SmallGroup(2000,250)<br />
|-<br />
|-[[M(125,3,19)]] || <math>k((C_5 \times C_5):SL_2(5).2)</math> || 1 || 20 || 2 || 10 || <math>C_4 \times C_2</math> || Inertial quotient as in M(125,3,17)<br />
|-<br />
|[[M(125,3,20)]] || <math>k5_+^{1+2}:(C_4 \times C_2)</math> || 1 || 14 || 8 ||8 || <math>C_4 \times C_2</math> || || || SmallGroup(1000,91)<br />
|-<br />
|[[M(125,3,21)]] || Faithful block of <math>k5_+^{1+2}:M_4(2)</math>, in which <math>M_4(2)'</math> acts trivially || 1 || 8 || 8 ||2 || <math>C_2 \times C_2</math> || || || SmallGroup(2000,264)<br />
|-<br />
|[[M(125,3,22)]] || <math>k5_+^{1+2}:D_8</math> || 1 || 14 || 8 ||5 || <math>D_8</math> || || || SmallGroup(1000,92)<br />
|-<br />
|[[M(125,3,23)]] || Faithful block of <math>k5_+^{1+2}:D_{16}</math>, in which <math>Z(D_{16})</math> acts trivially || 1 || 11 || 8 ||2 || <math>D_8</math> || || || <br />
|-<br />
|[[M(125,3,24)]] || <math>k5_+^{1+2}:Q_8</math> || 1 || 8 || 20 ||5 || <math>Q_8</math> || || || SmallGroup(1000,93)<br />
|-<br />
|[[M(125,3,25)]] || <math>k5_+^{1+2}:(C_3:C_4)</math> || 1 || 8 || 24 ||6 || <math>C_3:C_4</math> || || || SmallGroup(1500,35)<br />
|-<br />
|[[M(125,3,26)]] || <math>k5_+^{1+2}:C_{12}</math> || 1 || 14 || 12 ||12 || <math>C_{12}</math> || || || SmallGroup(1500,36)<br />
|-<br />
|[[M(125,3,27)]] || <math>k5_+^{1+2}:D_{12}</math> || 1 || 14 || 12 ||6 || <math>D_{12}</math> || || || SmallGroup(1000,37)<br />
|-<br />
|[[M(125,3,28)]] || Faithful block of <math>k5_+^{1+2}:D_{24}</math>, in which <math>Z(D_{24})</math> acts trivially || 1 || 10 || 4 ||1 || <math>D_{12}</math> || || || <br />
|-<br />
|[[M(125,3,29)]] || <math>k5_+^{1+2}:(C_4 \times C_4)</math> || 1 || 25 || 4 ||16 || <math>C_4 \times C_4</math> || || || SmallGroup(2000,473)<br />
|-<br />
|[[M(125,3,30)]] || Faithful block of <math>k5_+^{1+2}:(C_2.(C_4 \times C_4))</math>, in which <math>(C_2.(C_4 \times C_4))'</math> acts trivially || 1 || 13 || 4 ||4 || <math>C_4 \times C_4</math> || || || <math>C_2.(C_4 \times C_4</math> is SmallGroup(32,2)<br />
|-<br />
|[[M(125,3,31)]] || Faithful block of <math>k5_+^{1+2}:((C_4 \times C_4):C_4)</math>, in which <math>Z((C_4 \times C_4):C_4)</math> acts trivially || 1 || 10 || 4 ||1 || <math>C_4 \times C_4</math> || || || <math>(C_4 \times C_4):C_4</math> is SmallGroup(64,18)<br />
|-<br />
|[[M(125,3,32)]] || <math>k(C_5 \times C_5):GL_2(5)</math> || 1 || 25 || 4 ||20 || <math>C_4 \times C_4</math> || || || <br />
|-<br />
|[[M(125,3,33)]] || <math>B_0(kSL_3(5))</math> || 1 || 25 || 4 ||24 || <math>C_4 \times C_4</math> || || || <br />
|}</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Labelling_for_Morita_equivalence_classes&diff=1218
Labelling for Morita equivalence classes
2023-10-03T10:08:04Z
<p>Charles Eaton: /* Conventions */</p>
<hr />
<div>We use two compatible systems for labelling Morita equivalence classes of blocks of finite groups with respect to an algebraically closed field <math>k</math>. <br />
<br />
=== When the defect group is listed in the GAP SmallGroup library === <br />
<br />
M(x,y,z) is a class consisting of blocks with defect groups of order x, with a representative having defect group SmallGroup(x,y) in GAP/MAGMA labelling. It is the z-th such class. <br />
<br />
Note that it is not known that the isomorphism class of a defect group is a Morita invariant, so it could be that M(x,y1,z1)=M(x,y2,z2) for some <math>(y1,z1) \neq (y2,z2)</math>.<br />
<br />
An excellent resource for the SmallGroup library is the [https://people.maths.bris.ac.uk/~matyd/GroupNames/ Groupnames site].<br />
<br />
=== Alternative notation ===<br />
<br />
When the defect group is not listed in the SmallGroup library or is a generic group <math>P</math>, then we use the notation M(<math>P</math>,z), the z-th class of blocks with defect groups isomorphic to <math>P</math>. When <math>P</math> is cyclic we also use the notation M(<math>|P|</math>,1,z) since cyclic <math>p</math>-groups are labelled 1 in the SmallGroup library.<br />
<br />
=== <math>\mathcal{O}</math>-Morita equivalence classes ===<br />
<br />
At present there is no known example of a <math>k</math>-Morita equivalence class of blocks which splits into more than one Morita equivalence class with respect to a complete discrete valuation ring. If such an example arises, then we will bring in more notation for classes with respect to <math>\mathcal{O}</math><br />
<br />
=== Incomplete classifications ===<br />
<br />
When the classification for a given p-group is incomplete, this should be labelled clearly on the group's page, preferrably using the following:<br />
<br />
'''<pre style="color: red">CLASSIFICATION INCOMPLETE</pre>'''<br />
Until the classification is known to be complete the labelling is provisional. After completion the labelling may be adjusted to conform to the usual conventions.<br />
<br />
In cases where there are potential classes for which there is no known example (such as for blocks with cyclic defect group where there may be Brauer trees with no known block representative) these may be included as a row in the table of classes for that defect group, but with no label attached.<br />
<br />
=== Conventions ===<br />
<br />
There can be no canonical choice of labelling, but where possible please try to keep the following conventions in mind.<br />
<br />
*Try to follow existing classifications for labelling where possible, provided that they have a logical basis.<br />
<br />
*Try to label following increasing size of inertial quotient.<br />
<br />
*Where there is a relationship between classes for blocks with different defect groups, try to arrange the labellings accordingly.<br />
<br />
Once established, only change a labelling in exceptional circumstances and be very careful to make the change in every effected place. Make it clear in the title of the edit precisely what changes you are making.</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Labelling_for_Morita_equivalence_classes&diff=1217
Labelling for Morita equivalence classes
2023-10-03T09:57:24Z
<p>Charles Eaton: </p>
<hr />
<div>We use two compatible systems for labelling Morita equivalence classes of blocks of finite groups with respect to an algebraically closed field <math>k</math>. <br />
<br />
=== When the defect group is listed in the GAP SmallGroup library === <br />
<br />
M(x,y,z) is a class consisting of blocks with defect groups of order x, with a representative having defect group SmallGroup(x,y) in GAP/MAGMA labelling. It is the z-th such class. <br />
<br />
Note that it is not known that the isomorphism class of a defect group is a Morita invariant, so it could be that M(x,y1,z1)=M(x,y2,z2) for some <math>(y1,z1) \neq (y2,z2)</math>.<br />
<br />
An excellent resource for the SmallGroup library is the [https://people.maths.bris.ac.uk/~matyd/GroupNames/ Groupnames site].<br />
<br />
=== Alternative notation ===<br />
<br />
When the defect group is not listed in the SmallGroup library or is a generic group <math>P</math>, then we use the notation M(<math>P</math>,z), the z-th class of blocks with defect groups isomorphic to <math>P</math>. When <math>P</math> is cyclic we also use the notation M(<math>|P|</math>,1,z) since cyclic <math>p</math>-groups are labelled 1 in the SmallGroup library.<br />
<br />
=== <math>\mathcal{O}</math>-Morita equivalence classes ===<br />
<br />
At present there is no known example of a <math>k</math>-Morita equivalence class of blocks which splits into more than one Morita equivalence class with respect to a complete discrete valuation ring. If such an example arises, then we will bring in more notation for classes with respect to <math>\mathcal{O}</math><br />
<br />
=== Incomplete classifications ===<br />
<br />
When the classification for a given p-group is incomplete, this should be labelled clearly on the group's page, preferrably using the following:<br />
<br />
'''<pre style="color: red">CLASSIFICATION INCOMPLETE</pre>'''<br />
Until the classification is known to be complete the labelling is provisional. After completion the labelling may be adjusted to conform to the usual conventions.<br />
<br />
In cases where there are potential classes for which there is no known example (such as for blocks with cyclic defect group where there may be Brauer trees with no known block representative) these may be included as a row in the table of classes for that defect group, but with no label attached.<br />
<br />
=== Conventions ===<br />
<br />
There can be no canonical choice of labelling, but where possible please try to keep the following conventions in mind.<br />
<br />
*Try to follow existing classifications for labelling where possible, provided that they have a logical basis.<br />
<br />
*In general classifications should follow increasing number of simple modules.<br />
<br />
*Where there is a relationship between classes for blocks with different defect groups, try to arrange the labellings accordingly.<br />
<br />
Once established, only change a labelling in exceptional circumstances and be very careful to make the change in every effected place. Make it clear in the title of the edit precisely what changes you are making.</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Classification_by_p-group&diff=1216
Classification by p-group
2023-10-03T09:46:22Z
<p>Charles Eaton: /* Blocks for p=5 */</p>
<hr />
<div>'''Classification of Morita equivalences for blocks with a given defect group'''<br />
<br />
On this page we list classifications of Morita equivalence classes for each isomorphism class of p-groups in turn. [[Generic classifications by p-group class|Generic classifications for classes of p-groups can be found here]].<br />
<br />
See [[Labelling for Morita equivalence classes|this page]] for a description of the labelling conventions.<br />
<br />
== Blocks for <math> p=2 </math> ==<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>1 \leq |D| \leq 8</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|- <br />
| 2 || [[C2|1]] || [[C2|<math>C_2</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
| 4 || [[C4|1]] || [[C4|<math>C_4</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
| 4 || [[C2xC2|2]] || [[C2xC2|<math>C_2 \times C_2</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Er82], [Li94] ]] ||<br />
|- <br />
|8 || [[C8|1]] || [[C8|<math>C_8</math>]] ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|8 || [[C4xC2|2]] || [[C4xC2|<math>C_4 \times C_2</math>]] ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|8 || [[D8|3]] || [[D8|<math>D_8</math>]] ||6(?) || <math>k</math> || <math>k</math> || [[References|[Er87] ]] || Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] <br />
|-<br />
|8 || [[Q8|4]] || [[Q8|<math>Q_8</math>]] ||3(3) || <math>\mathcal{O}</math> || <math>k</math> || [[References|[Er88a], [Er88b], [HKL07], [Ei16]]] || <br />
|-<br />
|8 || [[C2xC2xC2|5]] || [[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]] || 8(8) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References| [Ea16]]] || Uses CFSG<br />
|}<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>|D|=16</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Donovan (w.r.t)?<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|16 || [[C16|1]] || [[C16|<math>C_{16}</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|16 || [[C4xC4|2]] || [[C4xC4|<math>C_4 \times C_4</math>]] || <math>\mathcal{O}</math> || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[EKKS14] ]] ||<br />
|-<br />
|16 || [[MNA(2,1)|3]] || [[MNA(2,1)]] || No || 3(?) || No || || [[References|[Sa11] ]] || Block invariants known<br />
|-<br />
|16 || [[C4:C4|4]] || [[C4:C4|<math>C_4:C_4</math>]] || <math>\mathcal{O}</math>|| 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[CG12], [Sa12b]]] ||<br />
|-<br />
|16 || [[C8xC2|5]] || [[C8xC2|<math>C_8 \times C_2</math>]]|| <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|16 || [[M16|6]] || [[M16|<math>M_{16}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[CG12], [Sa12b] ]] || <br />
|-<br />
|16 || [[D16|7]] || [[D16|<math>D_{16}</math>]] || <math>k</math>|| 5(?) || <math>k</math> || <math>k</math> || [[References|[Er87]]] || Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] <br />
|-<br />
|16 || [[SD16|8]] || [[SD16|<math>SD_{16}</math>]] || <math>k</math> || 7(?) || || || [[References|[Er88c], [Er90b]]] || Two other possible classes<br />
|-<br />
|16 || [[Q16|9]] || [[Q16|<math>Q_{16}</math>]] || No || 6(?) || || <math>k</math> || [[References|[Er88a], [Er88b], [Ho97]]] || Two possibly infinite families when <math>l(B)=2</math>. Classified over <math>\mathcal{O}</math> when <math>l(B)=3</math> in [[References#E|[Ei16]]]. Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20b]]]<br />
|-<br />
|16 || [[C4xC2xC2|10]] || [[C4xC2xC2|<math>C_4 \times C_2 \times C_2</math>]]|| <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[EL18a]]] ||<br />
|-<br />
|16 || [[D8xC2|11]] || [[D8xC2|<math>D_8 \times C_2</math>]] || No || || || || [[References|[Sa12] ]] || Block invariants known<br />
|-<br />
|16 || [[Q8xC2|12]] || [[Q8xC2|<math>Q_8 \times C_2</math>]] || <math>\mathcal{O}</math> || 3(3) || No || || [[References#E|[EL20]]] || Block invariants known by [[References#S|[Sa13]]]<br />
|-<br />
|16 || [[D8*C4|13]] || [[D8*C4|<math>D_8*C_4</math>]] || No || 3(?) || No || || [[References|[Sa13b] ]] || Block invariants known<br />
|-<br />
|16 || [[(C2)^4|14]] || [[(C2)^4|<math>(C_2)^4</math>]] || <math>\mathcal{O}</math> || 16(16) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Ea18] ]] ||<br />
|}<br />
<br />
The table for defect groups of order 32 takes as its starting point Table 13.1 of Sambale's book [[References|[Sa14]]].<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>|D|=32</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Donovan (w.r.t)?<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|32 || [[C32|1]] || [[C32|<math>C_{32}</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|32 || [[MNA(2,2)|2]] || [[MNA(2,2)|<math>MNA(2,2)</math>]] || <math>\mathcal{O}</math> || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EKS12]]] ||<br />
|-<br />
|32 || [[C8xC4|3]] || [[C8xC4|<math>C_8 \times C_4</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|-<br />
|32 || [[C8:C4|4]] || [[C8:C4|<math>C_8:C_4</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[MNA(3,1)|5]] || [[MNA(3,1)|<math>MNA(3,1)</math>]] || No || || || || [[References#S|[Sa11] ]] || Invariants known<br />
|-<br />
|32 || [[MNA(2,1):C2|6]] || [[MNA(3,1):C2|<math>MNA(2,1):C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,7)|7]] || [[SmallGroup(32,7)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] || <math>M_{16}:C_2</math><br />
|-<br />
|32 || [[2.MNA(2,1)|8]] || [[2.MNA(2,1)|<math>2.MNA(2,1)</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[D8:C4|9]] || [[D8:C4|<math>D_8:C_4</math>]] || No || || || || [[References#S|[Sa14,10.23]]] || Invariants known<br />
|-<br />
|32 || [[Q8:C4|10]] || [[Q8:C4|<math>Q_8:C_4</math>]] || No || || || || [[References#S|[Sa14,10.25]]] || Invariants known<br />
|-<br />
|32 || [[C4wrC2|11]] || [[C4wrC2|<math>C_4 \wr C_2</math>]] || No || || || || [[References#K|[Ku80]]] || Invariants known<br />
|-<br />
|32 || [[C4:C8|12]] || [[C4:C8|<math>C_4:C_8</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[C8:C4a|13]] || [[C8:C4a|<math>C_8:C_4=\langle a,b|a^8=b^4=1, ba=a^3b \rangle</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[C8:C4b|14]] || [[C8:C4b|<math>C_8:C_4=\langle a,b|a^8=b^4=1, ba=a^7b \rangle</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,15)|15]] || [[SmallGroup(32,15)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[C16xC2|16]] || [[C16xC2|<math>C_{16} \times C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|32 || [[M32|17]] || [[M32|<math>M_{32}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[D32|18]] || [[D32|<math>D_{32}</math>]] || <math>k</math> || 5(?) || <math>k</math> || <math>k</math> || [[References#E|[Er87]]] || Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] <br />
|-<br />
|32 || [[SD32|19]] || [[SD32|<math>SD_{32}</math>]] || <math>k</math> || || || || ||<br />
|-<br />
|32 || [[Q32|20]] || [[Q32|<math>Q_{32}</math>]] || No || || || || [[References#E|[Er88a], [Er88b], [Ho97]]] || Two possibly infinite families when <math>l(B)=2</math>. Classified over <math>\mathcal{O}</math> when <math>l(B)=3</math> in [[References#E|[Ei16]]]. Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20b]]]<br />
|-<br />
|32 || [[C4xC4xC2|21]] || [[C4xC4xC2|<math>C_4 \times C_4 \times C_2</math>]] || <math>\mathcal{O}</math> || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EKKS14]]] <br />
|-<br />
|32 || [[MNA(2,1)xC2|22]] || [[MNA(2,1)xC2|<math>MNA(2,1) \times C_2</math>]] || No || || || || [[References#S|[Sa14,10.25]]] || Invariants known<br />
|-<br />
|32 || [[(C4:C4)xC2|23]] || [[(C4:C4)xC2|<math>(C_4:C_4) \times C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,24)|24]] || [[SmallGroup(32,24)]]<!--<math>(C_4 \times C_4):C_2=\langle a,b,c \mid a^4 = b^4 = c^2 = e, ab = ba, ac = ca, cb = a^2bc \rangle</math>]]--> || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[D8xC4|25]] || [[D8xC4|<math>D_8 \times C_4</math>]] || No || || || || [[References#S|[Sa14,9.7]]] ||<br />
Invariants known<br />
|-<br />
|32 || [[Q8xC4|26]] || [[Q8xC4|<math>Q_8 \times C_4</math>]] || <math>\mathcal{O}</math> || 3(3) || No || || [[References#E|[EL20]]] || Invariants known by [[References#S|[Sa14,9.28]]]<br />
|-<br />
|32 || [[SmallGroup(32,27)|27]] || [[SmallGroup(32,27)]]<!--|<math>(C_4 \times C_4):C_2=\langle x,y,z,a,b \mid x^2 = y^2 = z^2 = a^2 = b^2 = e, xy = yx, xz, = zx, yz = zy, aza^{-1} = xz, bzb^{-1} = yz, ax = xa, ay = ya, bx = xb, by = yb \rangle</math>]]--> || No || || || || ||<br />
|-<br />
|32 || [[SmallGroup(32,28)|28]] || [[SmallGroup(32,28)]] || No || || || || [[References#S|[Sa14,13.11]]] || Invariants known<br />
|-<br />
|32 || [[SmallGroup(32,29)|29]] || [[SmallGroup(32,29)]] || No || || || || [[References#S|[Sa14,13.11]]] || Invariants known<br />
|-<br />
|32 || [[SmallGroup(32,30)|30]] || [[SmallGroup(32,30)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,31)|31]] || [[SmallGroup(32,31)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,32)|32]] || [[SmallGroup(32,32)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,33)|33]] || [[SmallGroup(32,33)]] || No || || || || [[References|[Sa14,13.12]]] || Invariants partly known<br />
|-<br />
|32 || [[SmallGroup(32,34)|34]] || [[SmallGroup(32,34)]] || No || || || || [[References|[Sa14,13.12]]] || Invariants partly known<br />
|-<br />
|32 || [[C4:Q8|35]] || [[C4:Q8|<math>C_4:Q_8</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[C8xC2xC2|36]] || [[C8xC2xC2|<math>C_8 \times C_2 \times C_2</math>]] || <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EL18a]]] ||<br />
|-<br />
|32 || [[M16xC2|37]] || [[M16xC2|<math>M_{16} \times C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[D8*C8|38]] || [[D8*C8|<math>D_8*C_8</math>]] || No || || || || [[References#S|[Sa14,9.18]]] || Invariants known<br />
|-<br />
|32 || [[D16xC2|39]] || [[D16xC2|<math>D_{16} \times C_2</math>]] || No || || || || [[References#S|[Sa14,9.7]]] || Invariants known<br />
|-<br />
|32 || [[SD16xC2|40]] || [[SD16xC2|<math>SD_{16} \times C_2</math>]] || No || || || || [[References#S|[Sa14,9.37]]] || Invariants known<br />
|-<br />
|32 || [[Q16xC2|41]] || [[Q16xC2|<math>Q_{16} \times C_2</math>]] || No || || || || [[References#S|[Sa14,9.28]]] || Invariants known<br />
|-<br />
|32 || [[D16*C4|42]] || [[D16*C4|<math>D_{16}*C_4</math>]] || No || || || || [[References|[Sa14,9.18]]] || Invariants known<br />
|-<br />
|32 || [[SmallGroup(32,43)|43]] || [[SmallGroup(32,43)]] || No || || || || ||<br />
|-<br />
|32 || [[SmallGroup(32,44)|44]] || [[SmallGroup(32,44)]] || No || || || || ||<br />
|-<br />
|32 || [[C4xC2xC2xC2|45]] || [[C4xC2xC2xC2|<math>C_4 \times C_2 \times C_2 \times C_2</math>]] || <math>\mathcal{O}</math> || || || || [[References#S|[Sa14, 13.9]]] || Invariants known<br />
|-<br />
|32 || [[D8xC2xC2|46]] || [[D8xC2xC2|<math>D_8 \times C_2 \times C_2</math>]] || No || || || || ||<br />
|-<br />
|32 || [[Q8xC2xC2|47]] || [[Q8xC2xC2|<math>Q_8 \times C_2 \times C_2</math>]] || No || || || || ||<br />
|-<br />
|32 || [[D8*C4xC2|48]] || [[D8*C4xC2|<math>(D_8*C_4) \times C_2</math>]] || No || || || || ||<br />
|-<br />
|32 || [[D8*D8|49]] || [[D8*D8|<math>D_8*D_8</math>]] || No || || || || [[References#S|[Sa13c]]] || Invariants partly known<br />
|-<br />
|32 || [[D8*Q8|50]] || [[D8*Q8|<math>D_8*Q_8</math>]] || No || || || || [[References#S|[Sa13c]]] || Invariants partly known<br />
|-<br />
|32 || [[(C2)^5|51]] || [[(C2)^5|<math>(C_2)^5</math>]] || <math>\mathcal{O}</math> || 34 (34) || <math>\mathcal{O}</math> || || [[References#A|[Ar19]]] || Derived eq. classes determined for 30 of the 34 Morita eq. classes. <br />
|}<br />
<br />
<!--<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>|D|=64</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Donovan (w.r.t)?<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|64 || [[C64|1]] || [[C64|<math>C_{64}</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
|64 || [[C8xC8|2]] || [[C8xC8|<math>C_8 \times C_8</math>]] || <math>\mathcal{O}</math> ||2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EKKS14]]]||<br />
|-<br />
|64 || [[SmallGroup(64,3)|3]] || [[SmallGroup(64,3)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_8:C_8</math><br />
|-<br />
|64 || [[(C2xC2xC2):C8|4]] || [[(C2xC2xC2):C8|<math>(C_2)^3:C_8</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,5)|5]] || [[SmallGroup(64,5)]] || No || || || || ||<br />
|-<br />
|64 || [[(D8:C8|6]] || [[D8:C8|<math>D_8:C_8</math>]] || No || || || || ||<br />
|-<br />
|64 || [[(Q8:C8|7]] || [[Q8:C8|<math>Q_8:C_8</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,8)|8]] || [[SmallGroup(64,8)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,9)|9]] || [[SmallGroup(64,9)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,10)|10]] || [[SmallGroup(64,10)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,11)|11]] || [[SmallGroup(64,11)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,12)|12]] || [[SmallGroup(64,12)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,13)|13]] || [[SmallGroup(64,13)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,14)|14]] || [[SmallGroup(64,14)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,15)|15]] || [[SmallGroup(64,15)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_8:C_8</math><br />
|-<br />
|64 || [[SmallGroup(64,16)|16]] || [[SmallGroup(64,16)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_8:C_8</math><br />
|-<br />
|64 || [[SmallGroup(64,17)|17]] || [[SmallGroup(64,17)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,18)|18]] || [[SmallGroup(64,18)]] || No || || || || || <math>(C_4 \times C_4):C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,19)|19]] || [[SmallGroup(64,19)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,20)|20]] || [[SmallGroup(64,20)]] || No || || || || || <math>(C_4 \times C_4):C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,21)|21]] || [[SmallGroup(64,21)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,22)|22]] || [[SmallGroup(64,22)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,23)|23]] || [[SmallGroup(64,23)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,24)|24]] || [[SmallGroup(64,24)]] || No || || || || || <math>M_{16}:C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,25)|25]] || [[SmallGroup(64,25)]] || No || || || || || <math>M_{16}:C_4</math><br />
|-<br />
|64 || [[C16xC4|26]] || [[C16xC4|<math>C_{16} \times C_4</math>]]|| <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|64 || [[SmallGroup(64,27)|27]] || [[SmallGroup(64,27)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_{16}:C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,28)|28]] || [[SmallGroup(64,28)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_{16}:C_4</math><br />
|-<br />
|64 || [[(C2xC2):C16|29]] || [[(C2xC2):C16|<math>(C_2)^2:C_{16}</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,30)|30]] || [[SmallGroup(64,30)]] || No || || || || || <math>M_{32}:C_2</math><br />
|-<br />
|64 || [[SmallGroup(64,31)|31]] || [[SmallGroup(64,31)]] || No || || || || || <math>M_{32}:C_2</math><br />
|-<br />
|64 || [[C2wrC4|32]] || [[(C2wrC4|<math>C_2 \wr C_4</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,33)|33]] || [[SmallGroup(64,33)]] || No || || || || || <br />
|-<br />
|64 || [[SmallGroup(64,34)|34]] || [[SmallGroup(64,31)]] || No || || || || || <math>(C_4 \times C_4):C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,35)|35]] || [[SmallGroup(64,35)]] || No || || || || || <math>(C_4 \times C_4):C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,36)|36]] || [[SmallGroup(64,36)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,37)|37]] || [[SmallGroup(64,37)]] || No || || || || || <math>C_4:Q_8</math>, fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,38)|38]] || [[SmallGroup(64,38)]] || No || || || || || <math>D_{16}:C_4</math>, fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,39)|39]] || [[SmallGroup(64,39)]] || No || || || || || <math>Q_{16}:C_2</math><br />
|-<br />
|64 || [[SmallGroup(64,40)|40]] || [[SmallGroup(64,40)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,79)|79]] || [[SmallGroup(64,79)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|-<br />
|64 || [[SmallGroup(64,81)|81]] || [[SmallGroup(64,81)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|}<br />
--><br />
<br />
==Blocks for <math>p=3</math>==<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>1 \leq |D| \leq 27</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|- <br />
| 3 || [[C3|1]] || [[C3|<math>C_3</math>]] || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
|9 || [[C9|1]] ||[[C9|<math>C_9</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|-<br />
|9 || [[C3xC3|2]] || [[C3xC3|<math>C_3 \times C_3</math>]] || || || || ||<br />
|-<br />
|27 || [[C27|1]] || [[C27|<math>C_{27}</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|27 || [[C9xC3|2]] || [[C9xC3|<math>C_9 \times C_3</math>]] || || || || ||<br />
|-<br />
|27 || [[3_+^3|3]] || [[3_+^3|<math>3_+^{1+2}</math>]] || || || || ||<br />
|-<br />
|27 || [[3_-^3|4]] || [[3_-^3|<math>3_-^{1+2}</math>]] || || || || ||<br />
|-<br />
|27 || [[C3xC3xC3|5]] || [[C3xC3xC3|<math>C_3 \times C_3 \times C_3</math>]] || || || || ||<br />
|}<br />
<br />
==Blocks for <math>p=5</math>==<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>5 \leq |D| \leq 125</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|- <br />
|5 || [[C5|1]] || [[C5|<math>C_5</math>]] ||6(6) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
|25 || [[C25|1]] ||[[C25|<math>C_{25}</math>]] || 6(6) || No || <math>\mathcal{O}</math> || || Max 12 classes <br />
|-<br />
|25 || [[C5xC5|2]] || [[C5xC5|<math>C_5 \times C_5</math>]] || || || || ||<br />
|- <br />
|125 || [[C125|1]] ||[[C125|<math>C_{125}</math>]] || || || || || <br />
|-<br />
|125 || [[C25xC5|2]] || [[C25xC5|<math>C_{25} \times C_5</math>]] || || || || || <br />
|-<br />
|125 || [[5_+^3|3]] || [[5_+^3|<math>5_+^{1+2}</math>]] || 62(62) || <math>\mathcal{O}</math> || || [[References#A|[AE23]]] || Inertial quotients are consistent within classes<br />
|-<br />
|125 || [[5_-^3|4]] || [[5_-^3|<math>5_-^{1+2}</math>]] || || || || ||<br />
|-<br />
|125 || [[C5xC5xC5|5]] || [[C5xC5xC5|<math>C_5 \times C_5 \times C_5</math>]] || || || || ||<br />
|}<br />
<br />
==Blocks for <math>p\geq 7</math>==<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>|D|</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|- <br />
|7 || [[C7|1]] || [[C7|<math>C_7</math>]] ||14(14) ||No || <math>\mathcal{O}</math> || ||Max 21 classes <br />
|- <br />
|11|| [[C11|1]] || [[C11|<math>C_{11}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|- <br />
|13 || [[C13|1]] || [[C13|<math>C_{13}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|-<br />
|17|| [[C17|1]] || [[C17|<math>C_{17}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|- <br />
|19 || [[C19|1]] || [[C19|<math>C_{19}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|-<br />
|23 || [[C23|1]] || [[C23|<math>C_{23}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|}</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Classification_by_p-group&diff=1215
Classification by p-group
2023-10-03T08:57:05Z
<p>Charles Eaton: /* Blocks for p=5 */</p>
<hr />
<div>'''Classification of Morita equivalences for blocks with a given defect group'''<br />
<br />
On this page we list classifications of Morita equivalence classes for each isomorphism class of p-groups in turn. [[Generic classifications by p-group class|Generic classifications for classes of p-groups can be found here]].<br />
<br />
See [[Labelling for Morita equivalence classes|this page]] for a description of the labelling conventions.<br />
<br />
== Blocks for <math> p=2 </math> ==<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>1 \leq |D| \leq 8</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|- <br />
| 2 || [[C2|1]] || [[C2|<math>C_2</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
| 4 || [[C4|1]] || [[C4|<math>C_4</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
| 4 || [[C2xC2|2]] || [[C2xC2|<math>C_2 \times C_2</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Er82], [Li94] ]] ||<br />
|- <br />
|8 || [[C8|1]] || [[C8|<math>C_8</math>]] ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|8 || [[C4xC2|2]] || [[C4xC2|<math>C_4 \times C_2</math>]] ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|8 || [[D8|3]] || [[D8|<math>D_8</math>]] ||6(?) || <math>k</math> || <math>k</math> || [[References|[Er87] ]] || Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] <br />
|-<br />
|8 || [[Q8|4]] || [[Q8|<math>Q_8</math>]] ||3(3) || <math>\mathcal{O}</math> || <math>k</math> || [[References|[Er88a], [Er88b], [HKL07], [Ei16]]] || <br />
|-<br />
|8 || [[C2xC2xC2|5]] || [[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]] || 8(8) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References| [Ea16]]] || Uses CFSG<br />
|}<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>|D|=16</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Donovan (w.r.t)?<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|16 || [[C16|1]] || [[C16|<math>C_{16}</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|16 || [[C4xC4|2]] || [[C4xC4|<math>C_4 \times C_4</math>]] || <math>\mathcal{O}</math> || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[EKKS14] ]] ||<br />
|-<br />
|16 || [[MNA(2,1)|3]] || [[MNA(2,1)]] || No || 3(?) || No || || [[References|[Sa11] ]] || Block invariants known<br />
|-<br />
|16 || [[C4:C4|4]] || [[C4:C4|<math>C_4:C_4</math>]] || <math>\mathcal{O}</math>|| 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[CG12], [Sa12b]]] ||<br />
|-<br />
|16 || [[C8xC2|5]] || [[C8xC2|<math>C_8 \times C_2</math>]]|| <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|16 || [[M16|6]] || [[M16|<math>M_{16}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[CG12], [Sa12b] ]] || <br />
|-<br />
|16 || [[D16|7]] || [[D16|<math>D_{16}</math>]] || <math>k</math>|| 5(?) || <math>k</math> || <math>k</math> || [[References|[Er87]]] || Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] <br />
|-<br />
|16 || [[SD16|8]] || [[SD16|<math>SD_{16}</math>]] || <math>k</math> || 7(?) || || || [[References|[Er88c], [Er90b]]] || Two other possible classes<br />
|-<br />
|16 || [[Q16|9]] || [[Q16|<math>Q_{16}</math>]] || No || 6(?) || || <math>k</math> || [[References|[Er88a], [Er88b], [Ho97]]] || Two possibly infinite families when <math>l(B)=2</math>. Classified over <math>\mathcal{O}</math> when <math>l(B)=3</math> in [[References#E|[Ei16]]]. Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20b]]]<br />
|-<br />
|16 || [[C4xC2xC2|10]] || [[C4xC2xC2|<math>C_4 \times C_2 \times C_2</math>]]|| <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[EL18a]]] ||<br />
|-<br />
|16 || [[D8xC2|11]] || [[D8xC2|<math>D_8 \times C_2</math>]] || No || || || || [[References|[Sa12] ]] || Block invariants known<br />
|-<br />
|16 || [[Q8xC2|12]] || [[Q8xC2|<math>Q_8 \times C_2</math>]] || <math>\mathcal{O}</math> || 3(3) || No || || [[References#E|[EL20]]] || Block invariants known by [[References#S|[Sa13]]]<br />
|-<br />
|16 || [[D8*C4|13]] || [[D8*C4|<math>D_8*C_4</math>]] || No || 3(?) || No || || [[References|[Sa13b] ]] || Block invariants known<br />
|-<br />
|16 || [[(C2)^4|14]] || [[(C2)^4|<math>(C_2)^4</math>]] || <math>\mathcal{O}</math> || 16(16) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Ea18] ]] ||<br />
|}<br />
<br />
The table for defect groups of order 32 takes as its starting point Table 13.1 of Sambale's book [[References|[Sa14]]].<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>|D|=32</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Donovan (w.r.t)?<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|32 || [[C32|1]] || [[C32|<math>C_{32}</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|32 || [[MNA(2,2)|2]] || [[MNA(2,2)|<math>MNA(2,2)</math>]] || <math>\mathcal{O}</math> || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EKS12]]] ||<br />
|-<br />
|32 || [[C8xC4|3]] || [[C8xC4|<math>C_8 \times C_4</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|-<br />
|32 || [[C8:C4|4]] || [[C8:C4|<math>C_8:C_4</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[MNA(3,1)|5]] || [[MNA(3,1)|<math>MNA(3,1)</math>]] || No || || || || [[References#S|[Sa11] ]] || Invariants known<br />
|-<br />
|32 || [[MNA(2,1):C2|6]] || [[MNA(3,1):C2|<math>MNA(2,1):C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,7)|7]] || [[SmallGroup(32,7)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] || <math>M_{16}:C_2</math><br />
|-<br />
|32 || [[2.MNA(2,1)|8]] || [[2.MNA(2,1)|<math>2.MNA(2,1)</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[D8:C4|9]] || [[D8:C4|<math>D_8:C_4</math>]] || No || || || || [[References#S|[Sa14,10.23]]] || Invariants known<br />
|-<br />
|32 || [[Q8:C4|10]] || [[Q8:C4|<math>Q_8:C_4</math>]] || No || || || || [[References#S|[Sa14,10.25]]] || Invariants known<br />
|-<br />
|32 || [[C4wrC2|11]] || [[C4wrC2|<math>C_4 \wr C_2</math>]] || No || || || || [[References#K|[Ku80]]] || Invariants known<br />
|-<br />
|32 || [[C4:C8|12]] || [[C4:C8|<math>C_4:C_8</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[C8:C4a|13]] || [[C8:C4a|<math>C_8:C_4=\langle a,b|a^8=b^4=1, ba=a^3b \rangle</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[C8:C4b|14]] || [[C8:C4b|<math>C_8:C_4=\langle a,b|a^8=b^4=1, ba=a^7b \rangle</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,15)|15]] || [[SmallGroup(32,15)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[C16xC2|16]] || [[C16xC2|<math>C_{16} \times C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|32 || [[M32|17]] || [[M32|<math>M_{32}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[D32|18]] || [[D32|<math>D_{32}</math>]] || <math>k</math> || 5(?) || <math>k</math> || <math>k</math> || [[References#E|[Er87]]] || Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] <br />
|-<br />
|32 || [[SD32|19]] || [[SD32|<math>SD_{32}</math>]] || <math>k</math> || || || || ||<br />
|-<br />
|32 || [[Q32|20]] || [[Q32|<math>Q_{32}</math>]] || No || || || || [[References#E|[Er88a], [Er88b], [Ho97]]] || Two possibly infinite families when <math>l(B)=2</math>. Classified over <math>\mathcal{O}</math> when <math>l(B)=3</math> in [[References#E|[Ei16]]]. Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20b]]]<br />
|-<br />
|32 || [[C4xC4xC2|21]] || [[C4xC4xC2|<math>C_4 \times C_4 \times C_2</math>]] || <math>\mathcal{O}</math> || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EKKS14]]] <br />
|-<br />
|32 || [[MNA(2,1)xC2|22]] || [[MNA(2,1)xC2|<math>MNA(2,1) \times C_2</math>]] || No || || || || [[References#S|[Sa14,10.25]]] || Invariants known<br />
|-<br />
|32 || [[(C4:C4)xC2|23]] || [[(C4:C4)xC2|<math>(C_4:C_4) \times C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,24)|24]] || [[SmallGroup(32,24)]]<!--<math>(C_4 \times C_4):C_2=\langle a,b,c \mid a^4 = b^4 = c^2 = e, ab = ba, ac = ca, cb = a^2bc \rangle</math>]]--> || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[D8xC4|25]] || [[D8xC4|<math>D_8 \times C_4</math>]] || No || || || || [[References#S|[Sa14,9.7]]] ||<br />
Invariants known<br />
|-<br />
|32 || [[Q8xC4|26]] || [[Q8xC4|<math>Q_8 \times C_4</math>]] || <math>\mathcal{O}</math> || 3(3) || No || || [[References#E|[EL20]]] || Invariants known by [[References#S|[Sa14,9.28]]]<br />
|-<br />
|32 || [[SmallGroup(32,27)|27]] || [[SmallGroup(32,27)]]<!--|<math>(C_4 \times C_4):C_2=\langle x,y,z,a,b \mid x^2 = y^2 = z^2 = a^2 = b^2 = e, xy = yx, xz, = zx, yz = zy, aza^{-1} = xz, bzb^{-1} = yz, ax = xa, ay = ya, bx = xb, by = yb \rangle</math>]]--> || No || || || || ||<br />
|-<br />
|32 || [[SmallGroup(32,28)|28]] || [[SmallGroup(32,28)]] || No || || || || [[References#S|[Sa14,13.11]]] || Invariants known<br />
|-<br />
|32 || [[SmallGroup(32,29)|29]] || [[SmallGroup(32,29)]] || No || || || || [[References#S|[Sa14,13.11]]] || Invariants known<br />
|-<br />
|32 || [[SmallGroup(32,30)|30]] || [[SmallGroup(32,30)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,31)|31]] || [[SmallGroup(32,31)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,32)|32]] || [[SmallGroup(32,32)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,33)|33]] || [[SmallGroup(32,33)]] || No || || || || [[References|[Sa14,13.12]]] || Invariants partly known<br />
|-<br />
|32 || [[SmallGroup(32,34)|34]] || [[SmallGroup(32,34)]] || No || || || || [[References|[Sa14,13.12]]] || Invariants partly known<br />
|-<br />
|32 || [[C4:Q8|35]] || [[C4:Q8|<math>C_4:Q_8</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[C8xC2xC2|36]] || [[C8xC2xC2|<math>C_8 \times C_2 \times C_2</math>]] || <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EL18a]]] ||<br />
|-<br />
|32 || [[M16xC2|37]] || [[M16xC2|<math>M_{16} \times C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[D8*C8|38]] || [[D8*C8|<math>D_8*C_8</math>]] || No || || || || [[References#S|[Sa14,9.18]]] || Invariants known<br />
|-<br />
|32 || [[D16xC2|39]] || [[D16xC2|<math>D_{16} \times C_2</math>]] || No || || || || [[References#S|[Sa14,9.7]]] || Invariants known<br />
|-<br />
|32 || [[SD16xC2|40]] || [[SD16xC2|<math>SD_{16} \times C_2</math>]] || No || || || || [[References#S|[Sa14,9.37]]] || Invariants known<br />
|-<br />
|32 || [[Q16xC2|41]] || [[Q16xC2|<math>Q_{16} \times C_2</math>]] || No || || || || [[References#S|[Sa14,9.28]]] || Invariants known<br />
|-<br />
|32 || [[D16*C4|42]] || [[D16*C4|<math>D_{16}*C_4</math>]] || No || || || || [[References|[Sa14,9.18]]] || Invariants known<br />
|-<br />
|32 || [[SmallGroup(32,43)|43]] || [[SmallGroup(32,43)]] || No || || || || ||<br />
|-<br />
|32 || [[SmallGroup(32,44)|44]] || [[SmallGroup(32,44)]] || No || || || || ||<br />
|-<br />
|32 || [[C4xC2xC2xC2|45]] || [[C4xC2xC2xC2|<math>C_4 \times C_2 \times C_2 \times C_2</math>]] || <math>\mathcal{O}</math> || || || || [[References#S|[Sa14, 13.9]]] || Invariants known<br />
|-<br />
|32 || [[D8xC2xC2|46]] || [[D8xC2xC2|<math>D_8 \times C_2 \times C_2</math>]] || No || || || || ||<br />
|-<br />
|32 || [[Q8xC2xC2|47]] || [[Q8xC2xC2|<math>Q_8 \times C_2 \times C_2</math>]] || No || || || || ||<br />
|-<br />
|32 || [[D8*C4xC2|48]] || [[D8*C4xC2|<math>(D_8*C_4) \times C_2</math>]] || No || || || || ||<br />
|-<br />
|32 || [[D8*D8|49]] || [[D8*D8|<math>D_8*D_8</math>]] || No || || || || [[References#S|[Sa13c]]] || Invariants partly known<br />
|-<br />
|32 || [[D8*Q8|50]] || [[D8*Q8|<math>D_8*Q_8</math>]] || No || || || || [[References#S|[Sa13c]]] || Invariants partly known<br />
|-<br />
|32 || [[(C2)^5|51]] || [[(C2)^5|<math>(C_2)^5</math>]] || <math>\mathcal{O}</math> || 34 (34) || <math>\mathcal{O}</math> || || [[References#A|[Ar19]]] || Derived eq. classes determined for 30 of the 34 Morita eq. classes. <br />
|}<br />
<br />
<!--<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>|D|=64</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Donovan (w.r.t)?<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|64 || [[C64|1]] || [[C64|<math>C_{64}</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
|64 || [[C8xC8|2]] || [[C8xC8|<math>C_8 \times C_8</math>]] || <math>\mathcal{O}</math> ||2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EKKS14]]]||<br />
|-<br />
|64 || [[SmallGroup(64,3)|3]] || [[SmallGroup(64,3)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_8:C_8</math><br />
|-<br />
|64 || [[(C2xC2xC2):C8|4]] || [[(C2xC2xC2):C8|<math>(C_2)^3:C_8</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,5)|5]] || [[SmallGroup(64,5)]] || No || || || || ||<br />
|-<br />
|64 || [[(D8:C8|6]] || [[D8:C8|<math>D_8:C_8</math>]] || No || || || || ||<br />
|-<br />
|64 || [[(Q8:C8|7]] || [[Q8:C8|<math>Q_8:C_8</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,8)|8]] || [[SmallGroup(64,8)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,9)|9]] || [[SmallGroup(64,9)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,10)|10]] || [[SmallGroup(64,10)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,11)|11]] || [[SmallGroup(64,11)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,12)|12]] || [[SmallGroup(64,12)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,13)|13]] || [[SmallGroup(64,13)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,14)|14]] || [[SmallGroup(64,14)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,15)|15]] || [[SmallGroup(64,15)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_8:C_8</math><br />
|-<br />
|64 || [[SmallGroup(64,16)|16]] || [[SmallGroup(64,16)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_8:C_8</math><br />
|-<br />
|64 || [[SmallGroup(64,17)|17]] || [[SmallGroup(64,17)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,18)|18]] || [[SmallGroup(64,18)]] || No || || || || || <math>(C_4 \times C_4):C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,19)|19]] || [[SmallGroup(64,19)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,20)|20]] || [[SmallGroup(64,20)]] || No || || || || || <math>(C_4 \times C_4):C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,21)|21]] || [[SmallGroup(64,21)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,22)|22]] || [[SmallGroup(64,22)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,23)|23]] || [[SmallGroup(64,23)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,24)|24]] || [[SmallGroup(64,24)]] || No || || || || || <math>M_{16}:C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,25)|25]] || [[SmallGroup(64,25)]] || No || || || || || <math>M_{16}:C_4</math><br />
|-<br />
|64 || [[C16xC4|26]] || [[C16xC4|<math>C_{16} \times C_4</math>]]|| <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|64 || [[SmallGroup(64,27)|27]] || [[SmallGroup(64,27)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_{16}:C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,28)|28]] || [[SmallGroup(64,28)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_{16}:C_4</math><br />
|-<br />
|64 || [[(C2xC2):C16|29]] || [[(C2xC2):C16|<math>(C_2)^2:C_{16}</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,30)|30]] || [[SmallGroup(64,30)]] || No || || || || || <math>M_{32}:C_2</math><br />
|-<br />
|64 || [[SmallGroup(64,31)|31]] || [[SmallGroup(64,31)]] || No || || || || || <math>M_{32}:C_2</math><br />
|-<br />
|64 || [[C2wrC4|32]] || [[(C2wrC4|<math>C_2 \wr C_4</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,33)|33]] || [[SmallGroup(64,33)]] || No || || || || || <br />
|-<br />
|64 || [[SmallGroup(64,34)|34]] || [[SmallGroup(64,31)]] || No || || || || || <math>(C_4 \times C_4):C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,35)|35]] || [[SmallGroup(64,35)]] || No || || || || || <math>(C_4 \times C_4):C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,36)|36]] || [[SmallGroup(64,36)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,37)|37]] || [[SmallGroup(64,37)]] || No || || || || || <math>C_4:Q_8</math>, fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,38)|38]] || [[SmallGroup(64,38)]] || No || || || || || <math>D_{16}:C_4</math>, fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,39)|39]] || [[SmallGroup(64,39)]] || No || || || || || <math>Q_{16}:C_2</math><br />
|-<br />
|64 || [[SmallGroup(64,40)|40]] || [[SmallGroup(64,40)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,79)|79]] || [[SmallGroup(64,79)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|-<br />
|64 || [[SmallGroup(64,81)|81]] || [[SmallGroup(64,81)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|}<br />
--><br />
<br />
==Blocks for <math>p=3</math>==<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>1 \leq |D| \leq 27</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|- <br />
| 3 || [[C3|1]] || [[C3|<math>C_3</math>]] || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
|9 || [[C9|1]] ||[[C9|<math>C_9</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|-<br />
|9 || [[C3xC3|2]] || [[C3xC3|<math>C_3 \times C_3</math>]] || || || || ||<br />
|-<br />
|27 || [[C27|1]] || [[C27|<math>C_{27}</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|27 || [[C9xC3|2]] || [[C9xC3|<math>C_9 \times C_3</math>]] || || || || ||<br />
|-<br />
|27 || [[3_+^3|3]] || [[3_+^3|<math>3_+^{1+2}</math>]] || || || || ||<br />
|-<br />
|27 || [[3_-^3|4]] || [[3_-^3|<math>3_-^{1+2}</math>]] || || || || ||<br />
|-<br />
|27 || [[C3xC3xC3|5]] || [[C3xC3xC3|<math>C_3 \times C_3 \times C_3</math>]] || || || || ||<br />
|}<br />
<br />
==Blocks for <math>p=5</math>==<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>5 \leq |D| \leq 125</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|- <br />
|5 || [[C5|1]] || [[C5|<math>C_5</math>]] ||6(6) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
|25 || [[C25|1]] ||[[C25|<math>C_{25}</math>]] || 6(6) || No || <math>\mathcal{O}</math> || || Max 12 classes <br />
|-<br />
|25 || [[C5xC5|2]] || [[C5xC5|<math>C_5 \times C_5</math>]] || || || || ||<br />
|- <br />
|125 || [[C125|1]] ||[[C125|<math>C_{125}</math>]] || || || || || <br />
|-<br />
|125 || [[C25xC5|2]] || [[C25xC5|<math>C_{25} \times C_5</math>]] || || || || || <br />
|-<br />
|125 || [[5_+^3|3]] || [[5_+^3|<math>5_+^{1+2}</math>]] || 62(62) || <math>\mathcal{O}</math> || || [[References#A|[AE23]]] ||<br />
|-<br />
|125 || [[5_-^3|4]] || [[5_-^3|<math>5_-^{1+2}</math>]] || || || || ||<br />
|-<br />
|125 || [[C5xC5xC5|5]] || [[C5xC5xC5|<math>C_5 \times C_5 \times C_5</math>]] || || || || ||<br />
|}<br />
<br />
==Blocks for <math>p\geq 7</math>==<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>|D|</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|- <br />
|7 || [[C7|1]] || [[C7|<math>C_7</math>]] ||14(14) ||No || <math>\mathcal{O}</math> || ||Max 21 classes <br />
|- <br />
|11|| [[C11|1]] || [[C11|<math>C_{11}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|- <br />
|13 || [[C13|1]] || [[C13|<math>C_{13}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|-<br />
|17|| [[C17|1]] || [[C17|<math>C_{17}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|- <br />
|19 || [[C19|1]] || [[C19|<math>C_{19}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|-<br />
|23 || [[C23|1]] || [[C23|<math>C_{23}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|}</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Classification_by_p-group&diff=1214
Classification by p-group
2023-10-03T08:56:07Z
<p>Charles Eaton: /* Blocks for p=5 */</p>
<hr />
<div>'''Classification of Morita equivalences for blocks with a given defect group'''<br />
<br />
On this page we list classifications of Morita equivalence classes for each isomorphism class of p-groups in turn. [[Generic classifications by p-group class|Generic classifications for classes of p-groups can be found here]].<br />
<br />
See [[Labelling for Morita equivalence classes|this page]] for a description of the labelling conventions.<br />
<br />
== Blocks for <math> p=2 </math> ==<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>1 \leq |D| \leq 8</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|- <br />
| 2 || [[C2|1]] || [[C2|<math>C_2</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
| 4 || [[C4|1]] || [[C4|<math>C_4</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
| 4 || [[C2xC2|2]] || [[C2xC2|<math>C_2 \times C_2</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Er82], [Li94] ]] ||<br />
|- <br />
|8 || [[C8|1]] || [[C8|<math>C_8</math>]] ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|8 || [[C4xC2|2]] || [[C4xC2|<math>C_4 \times C_2</math>]] ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|8 || [[D8|3]] || [[D8|<math>D_8</math>]] ||6(?) || <math>k</math> || <math>k</math> || [[References|[Er87] ]] || Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] <br />
|-<br />
|8 || [[Q8|4]] || [[Q8|<math>Q_8</math>]] ||3(3) || <math>\mathcal{O}</math> || <math>k</math> || [[References|[Er88a], [Er88b], [HKL07], [Ei16]]] || <br />
|-<br />
|8 || [[C2xC2xC2|5]] || [[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]] || 8(8) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References| [Ea16]]] || Uses CFSG<br />
|}<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>|D|=16</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Donovan (w.r.t)?<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|16 || [[C16|1]] || [[C16|<math>C_{16}</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|16 || [[C4xC4|2]] || [[C4xC4|<math>C_4 \times C_4</math>]] || <math>\mathcal{O}</math> || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[EKKS14] ]] ||<br />
|-<br />
|16 || [[MNA(2,1)|3]] || [[MNA(2,1)]] || No || 3(?) || No || || [[References|[Sa11] ]] || Block invariants known<br />
|-<br />
|16 || [[C4:C4|4]] || [[C4:C4|<math>C_4:C_4</math>]] || <math>\mathcal{O}</math>|| 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[CG12], [Sa12b]]] ||<br />
|-<br />
|16 || [[C8xC2|5]] || [[C8xC2|<math>C_8 \times C_2</math>]]|| <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|16 || [[M16|6]] || [[M16|<math>M_{16}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[CG12], [Sa12b] ]] || <br />
|-<br />
|16 || [[D16|7]] || [[D16|<math>D_{16}</math>]] || <math>k</math>|| 5(?) || <math>k</math> || <math>k</math> || [[References|[Er87]]] || Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] <br />
|-<br />
|16 || [[SD16|8]] || [[SD16|<math>SD_{16}</math>]] || <math>k</math> || 7(?) || || || [[References|[Er88c], [Er90b]]] || Two other possible classes<br />
|-<br />
|16 || [[Q16|9]] || [[Q16|<math>Q_{16}</math>]] || No || 6(?) || || <math>k</math> || [[References|[Er88a], [Er88b], [Ho97]]] || Two possibly infinite families when <math>l(B)=2</math>. Classified over <math>\mathcal{O}</math> when <math>l(B)=3</math> in [[References#E|[Ei16]]]. Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20b]]]<br />
|-<br />
|16 || [[C4xC2xC2|10]] || [[C4xC2xC2|<math>C_4 \times C_2 \times C_2</math>]]|| <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[EL18a]]] ||<br />
|-<br />
|16 || [[D8xC2|11]] || [[D8xC2|<math>D_8 \times C_2</math>]] || No || || || || [[References|[Sa12] ]] || Block invariants known<br />
|-<br />
|16 || [[Q8xC2|12]] || [[Q8xC2|<math>Q_8 \times C_2</math>]] || <math>\mathcal{O}</math> || 3(3) || No || || [[References#E|[EL20]]] || Block invariants known by [[References#S|[Sa13]]]<br />
|-<br />
|16 || [[D8*C4|13]] || [[D8*C4|<math>D_8*C_4</math>]] || No || 3(?) || No || || [[References|[Sa13b] ]] || Block invariants known<br />
|-<br />
|16 || [[(C2)^4|14]] || [[(C2)^4|<math>(C_2)^4</math>]] || <math>\mathcal{O}</math> || 16(16) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Ea18] ]] ||<br />
|}<br />
<br />
The table for defect groups of order 32 takes as its starting point Table 13.1 of Sambale's book [[References|[Sa14]]].<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>|D|=32</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Donovan (w.r.t)?<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|32 || [[C32|1]] || [[C32|<math>C_{32}</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|32 || [[MNA(2,2)|2]] || [[MNA(2,2)|<math>MNA(2,2)</math>]] || <math>\mathcal{O}</math> || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EKS12]]] ||<br />
|-<br />
|32 || [[C8xC4|3]] || [[C8xC4|<math>C_8 \times C_4</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|-<br />
|32 || [[C8:C4|4]] || [[C8:C4|<math>C_8:C_4</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[MNA(3,1)|5]] || [[MNA(3,1)|<math>MNA(3,1)</math>]] || No || || || || [[References#S|[Sa11] ]] || Invariants known<br />
|-<br />
|32 || [[MNA(2,1):C2|6]] || [[MNA(3,1):C2|<math>MNA(2,1):C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,7)|7]] || [[SmallGroup(32,7)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] || <math>M_{16}:C_2</math><br />
|-<br />
|32 || [[2.MNA(2,1)|8]] || [[2.MNA(2,1)|<math>2.MNA(2,1)</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[D8:C4|9]] || [[D8:C4|<math>D_8:C_4</math>]] || No || || || || [[References#S|[Sa14,10.23]]] || Invariants known<br />
|-<br />
|32 || [[Q8:C4|10]] || [[Q8:C4|<math>Q_8:C_4</math>]] || No || || || || [[References#S|[Sa14,10.25]]] || Invariants known<br />
|-<br />
|32 || [[C4wrC2|11]] || [[C4wrC2|<math>C_4 \wr C_2</math>]] || No || || || || [[References#K|[Ku80]]] || Invariants known<br />
|-<br />
|32 || [[C4:C8|12]] || [[C4:C8|<math>C_4:C_8</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[C8:C4a|13]] || [[C8:C4a|<math>C_8:C_4=\langle a,b|a^8=b^4=1, ba=a^3b \rangle</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[C8:C4b|14]] || [[C8:C4b|<math>C_8:C_4=\langle a,b|a^8=b^4=1, ba=a^7b \rangle</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,15)|15]] || [[SmallGroup(32,15)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[C16xC2|16]] || [[C16xC2|<math>C_{16} \times C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|32 || [[M32|17]] || [[M32|<math>M_{32}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[D32|18]] || [[D32|<math>D_{32}</math>]] || <math>k</math> || 5(?) || <math>k</math> || <math>k</math> || [[References#E|[Er87]]] || Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] <br />
|-<br />
|32 || [[SD32|19]] || [[SD32|<math>SD_{32}</math>]] || <math>k</math> || || || || ||<br />
|-<br />
|32 || [[Q32|20]] || [[Q32|<math>Q_{32}</math>]] || No || || || || [[References#E|[Er88a], [Er88b], [Ho97]]] || Two possibly infinite families when <math>l(B)=2</math>. Classified over <math>\mathcal{O}</math> when <math>l(B)=3</math> in [[References#E|[Ei16]]]. Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20b]]]<br />
|-<br />
|32 || [[C4xC4xC2|21]] || [[C4xC4xC2|<math>C_4 \times C_4 \times C_2</math>]] || <math>\mathcal{O}</math> || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EKKS14]]] <br />
|-<br />
|32 || [[MNA(2,1)xC2|22]] || [[MNA(2,1)xC2|<math>MNA(2,1) \times C_2</math>]] || No || || || || [[References#S|[Sa14,10.25]]] || Invariants known<br />
|-<br />
|32 || [[(C4:C4)xC2|23]] || [[(C4:C4)xC2|<math>(C_4:C_4) \times C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,24)|24]] || [[SmallGroup(32,24)]]<!--<math>(C_4 \times C_4):C_2=\langle a,b,c \mid a^4 = b^4 = c^2 = e, ab = ba, ac = ca, cb = a^2bc \rangle</math>]]--> || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[D8xC4|25]] || [[D8xC4|<math>D_8 \times C_4</math>]] || No || || || || [[References#S|[Sa14,9.7]]] ||<br />
Invariants known<br />
|-<br />
|32 || [[Q8xC4|26]] || [[Q8xC4|<math>Q_8 \times C_4</math>]] || <math>\mathcal{O}</math> || 3(3) || No || || [[References#E|[EL20]]] || Invariants known by [[References#S|[Sa14,9.28]]]<br />
|-<br />
|32 || [[SmallGroup(32,27)|27]] || [[SmallGroup(32,27)]]<!--|<math>(C_4 \times C_4):C_2=\langle x,y,z,a,b \mid x^2 = y^2 = z^2 = a^2 = b^2 = e, xy = yx, xz, = zx, yz = zy, aza^{-1} = xz, bzb^{-1} = yz, ax = xa, ay = ya, bx = xb, by = yb \rangle</math>]]--> || No || || || || ||<br />
|-<br />
|32 || [[SmallGroup(32,28)|28]] || [[SmallGroup(32,28)]] || No || || || || [[References#S|[Sa14,13.11]]] || Invariants known<br />
|-<br />
|32 || [[SmallGroup(32,29)|29]] || [[SmallGroup(32,29)]] || No || || || || [[References#S|[Sa14,13.11]]] || Invariants known<br />
|-<br />
|32 || [[SmallGroup(32,30)|30]] || [[SmallGroup(32,30)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,31)|31]] || [[SmallGroup(32,31)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,32)|32]] || [[SmallGroup(32,32)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,33)|33]] || [[SmallGroup(32,33)]] || No || || || || [[References|[Sa14,13.12]]] || Invariants partly known<br />
|-<br />
|32 || [[SmallGroup(32,34)|34]] || [[SmallGroup(32,34)]] || No || || || || [[References|[Sa14,13.12]]] || Invariants partly known<br />
|-<br />
|32 || [[C4:Q8|35]] || [[C4:Q8|<math>C_4:Q_8</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[C8xC2xC2|36]] || [[C8xC2xC2|<math>C_8 \times C_2 \times C_2</math>]] || <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EL18a]]] ||<br />
|-<br />
|32 || [[M16xC2|37]] || [[M16xC2|<math>M_{16} \times C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[D8*C8|38]] || [[D8*C8|<math>D_8*C_8</math>]] || No || || || || [[References#S|[Sa14,9.18]]] || Invariants known<br />
|-<br />
|32 || [[D16xC2|39]] || [[D16xC2|<math>D_{16} \times C_2</math>]] || No || || || || [[References#S|[Sa14,9.7]]] || Invariants known<br />
|-<br />
|32 || [[SD16xC2|40]] || [[SD16xC2|<math>SD_{16} \times C_2</math>]] || No || || || || [[References#S|[Sa14,9.37]]] || Invariants known<br />
|-<br />
|32 || [[Q16xC2|41]] || [[Q16xC2|<math>Q_{16} \times C_2</math>]] || No || || || || [[References#S|[Sa14,9.28]]] || Invariants known<br />
|-<br />
|32 || [[D16*C4|42]] || [[D16*C4|<math>D_{16}*C_4</math>]] || No || || || || [[References|[Sa14,9.18]]] || Invariants known<br />
|-<br />
|32 || [[SmallGroup(32,43)|43]] || [[SmallGroup(32,43)]] || No || || || || ||<br />
|-<br />
|32 || [[SmallGroup(32,44)|44]] || [[SmallGroup(32,44)]] || No || || || || ||<br />
|-<br />
|32 || [[C4xC2xC2xC2|45]] || [[C4xC2xC2xC2|<math>C_4 \times C_2 \times C_2 \times C_2</math>]] || <math>\mathcal{O}</math> || || || || [[References#S|[Sa14, 13.9]]] || Invariants known<br />
|-<br />
|32 || [[D8xC2xC2|46]] || [[D8xC2xC2|<math>D_8 \times C_2 \times C_2</math>]] || No || || || || ||<br />
|-<br />
|32 || [[Q8xC2xC2|47]] || [[Q8xC2xC2|<math>Q_8 \times C_2 \times C_2</math>]] || No || || || || ||<br />
|-<br />
|32 || [[D8*C4xC2|48]] || [[D8*C4xC2|<math>(D_8*C_4) \times C_2</math>]] || No || || || || ||<br />
|-<br />
|32 || [[D8*D8|49]] || [[D8*D8|<math>D_8*D_8</math>]] || No || || || || [[References#S|[Sa13c]]] || Invariants partly known<br />
|-<br />
|32 || [[D8*Q8|50]] || [[D8*Q8|<math>D_8*Q_8</math>]] || No || || || || [[References#S|[Sa13c]]] || Invariants partly known<br />
|-<br />
|32 || [[(C2)^5|51]] || [[(C2)^5|<math>(C_2)^5</math>]] || <math>\mathcal{O}</math> || 34 (34) || <math>\mathcal{O}</math> || || [[References#A|[Ar19]]] || Derived eq. classes determined for 30 of the 34 Morita eq. classes. <br />
|}<br />
<br />
<!--<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>|D|=64</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Donovan (w.r.t)?<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|64 || [[C64|1]] || [[C64|<math>C_{64}</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
|64 || [[C8xC8|2]] || [[C8xC8|<math>C_8 \times C_8</math>]] || <math>\mathcal{O}</math> ||2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EKKS14]]]||<br />
|-<br />
|64 || [[SmallGroup(64,3)|3]] || [[SmallGroup(64,3)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_8:C_8</math><br />
|-<br />
|64 || [[(C2xC2xC2):C8|4]] || [[(C2xC2xC2):C8|<math>(C_2)^3:C_8</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,5)|5]] || [[SmallGroup(64,5)]] || No || || || || ||<br />
|-<br />
|64 || [[(D8:C8|6]] || [[D8:C8|<math>D_8:C_8</math>]] || No || || || || ||<br />
|-<br />
|64 || [[(Q8:C8|7]] || [[Q8:C8|<math>Q_8:C_8</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,8)|8]] || [[SmallGroup(64,8)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,9)|9]] || [[SmallGroup(64,9)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,10)|10]] || [[SmallGroup(64,10)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,11)|11]] || [[SmallGroup(64,11)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,12)|12]] || [[SmallGroup(64,12)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,13)|13]] || [[SmallGroup(64,13)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,14)|14]] || [[SmallGroup(64,14)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,15)|15]] || [[SmallGroup(64,15)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_8:C_8</math><br />
|-<br />
|64 || [[SmallGroup(64,16)|16]] || [[SmallGroup(64,16)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_8:C_8</math><br />
|-<br />
|64 || [[SmallGroup(64,17)|17]] || [[SmallGroup(64,17)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,18)|18]] || [[SmallGroup(64,18)]] || No || || || || || <math>(C_4 \times C_4):C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,19)|19]] || [[SmallGroup(64,19)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,20)|20]] || [[SmallGroup(64,20)]] || No || || || || || <math>(C_4 \times C_4):C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,21)|21]] || [[SmallGroup(64,21)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,22)|22]] || [[SmallGroup(64,22)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,23)|23]] || [[SmallGroup(64,23)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,24)|24]] || [[SmallGroup(64,24)]] || No || || || || || <math>M_{16}:C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,25)|25]] || [[SmallGroup(64,25)]] || No || || || || || <math>M_{16}:C_4</math><br />
|-<br />
|64 || [[C16xC4|26]] || [[C16xC4|<math>C_{16} \times C_4</math>]]|| <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|64 || [[SmallGroup(64,27)|27]] || [[SmallGroup(64,27)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_{16}:C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,28)|28]] || [[SmallGroup(64,28)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_{16}:C_4</math><br />
|-<br />
|64 || [[(C2xC2):C16|29]] || [[(C2xC2):C16|<math>(C_2)^2:C_{16}</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,30)|30]] || [[SmallGroup(64,30)]] || No || || || || || <math>M_{32}:C_2</math><br />
|-<br />
|64 || [[SmallGroup(64,31)|31]] || [[SmallGroup(64,31)]] || No || || || || || <math>M_{32}:C_2</math><br />
|-<br />
|64 || [[C2wrC4|32]] || [[(C2wrC4|<math>C_2 \wr C_4</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,33)|33]] || [[SmallGroup(64,33)]] || No || || || || || <br />
|-<br />
|64 || [[SmallGroup(64,34)|34]] || [[SmallGroup(64,31)]] || No || || || || || <math>(C_4 \times C_4):C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,35)|35]] || [[SmallGroup(64,35)]] || No || || || || || <math>(C_4 \times C_4):C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,36)|36]] || [[SmallGroup(64,36)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,37)|37]] || [[SmallGroup(64,37)]] || No || || || || || <math>C_4:Q_8</math>, fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,38)|38]] || [[SmallGroup(64,38)]] || No || || || || || <math>D_{16}:C_4</math>, fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,39)|39]] || [[SmallGroup(64,39)]] || No || || || || || <math>Q_{16}:C_2</math><br />
|-<br />
|64 || [[SmallGroup(64,40)|40]] || [[SmallGroup(64,40)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,79)|79]] || [[SmallGroup(64,79)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|-<br />
|64 || [[SmallGroup(64,81)|81]] || [[SmallGroup(64,81)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|}<br />
--><br />
<br />
==Blocks for <math>p=3</math>==<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>1 \leq |D| \leq 27</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|- <br />
| 3 || [[C3|1]] || [[C3|<math>C_3</math>]] || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
|9 || [[C9|1]] ||[[C9|<math>C_9</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|-<br />
|9 || [[C3xC3|2]] || [[C3xC3|<math>C_3 \times C_3</math>]] || || || || ||<br />
|-<br />
|27 || [[C27|1]] || [[C27|<math>C_{27}</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|27 || [[C9xC3|2]] || [[C9xC3|<math>C_9 \times C_3</math>]] || || || || ||<br />
|-<br />
|27 || [[3_+^3|3]] || [[3_+^3|<math>3_+^{1+2}</math>]] || || || || ||<br />
|-<br />
|27 || [[3_-^3|4]] || [[3_-^3|<math>3_-^{1+2}</math>]] || || || || ||<br />
|-<br />
|27 || [[C3xC3xC3|5]] || [[C3xC3xC3|<math>C_3 \times C_3 \times C_3</math>]] || || || || ||<br />
|}<br />
<br />
==Blocks for <math>p=5</math>==<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>5 \leq |D| \leq 125</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|- <br />
|5 || [[C5|1]] || [[C5|<math>C_5</math>]] ||6(6) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
|25 || [[C25|1]] ||[[C25|<math>C_{25}</math>]] || 6(6) || No || <math>\mathcal{O}</math> || || Max 12 classes <br />
|-<br />
|25 || [[C5xC5|2]] || [[C5xC5|<math>C_5 \times C_5</math>]] || || || || ||<br />
|- <br />
|125 || [[C125|1]] ||[[C125|<math>C_{125}</math>]] || || || || || <br />
|-<br />
|125 || [[C25xC5|2]] || [[C25xC5|<math>C_{25} \times C_5</math>]] || || || || || <br />
|-<br />
|125 || [[5_+^3|3]] || [[5_+^3|<math>5_+^{1+2}</math>]] || 62(62) || <math>\mathcal{O}</math> || || [[Reference#A|[AE23]]] ||<br />
|-<br />
|125 || [[5_-^3|4]] || [[5_-^3|<math>5_-^{1+2}</math>]] || || || || ||<br />
|-<br />
|125 || [[C5xC5xC5|5]] || [[C5xC5xC5|<math>C_5 \times C_5 \times C_5</math> || || || || ||<br />
|}<br />
<br />
==Blocks for <math>p\geq 7</math>==<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>|D|</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|- <br />
|7 || [[C7|1]] || [[C7|<math>C_7</math>]] ||14(14) ||No || <math>\mathcal{O}</math> || ||Max 21 classes <br />
|- <br />
|11|| [[C11|1]] || [[C11|<math>C_{11}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|- <br />
|13 || [[C13|1]] || [[C13|<math>C_{13}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|-<br />
|17|| [[C17|1]] || [[C17|<math>C_{17}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|- <br />
|19 || [[C19|1]] || [[C19|<math>C_{19}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|-<br />
|23 || [[C23|1]] || [[C23|<math>C_{23}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|}</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Reductions&diff=1213
Reductions
2023-10-03T08:34:17Z
<p>Charles Eaton: </p>
<hr />
<div>This page will contain descriptions of reduction techniques and results.<br />
<br />
== Donovan's conjecture ==<br />
<br />
[[Statements of conjectures#Donovan's conjecture|For the statement of the conjecture click here.]]<br />
<br />
=== <math>k</math>-Donovan conjecture ===<br />
<br />
By [[References#K|[Kü95]]] it suffices to consider blocks of finite groups that are generated by the defect groups, i.e., the defect groups are contained in no proper normal subgroup.<br />
<br />
Several reductions were achieved in [[References#D|[Du04]]], but these have been subsumed in later work.<br />
<br />
'''<math>P</math> abelian:''' To show the <math>k</math>-Donovan conjecture for abelian <math>p</math>-groups, it suffices to verify the [[Statements of conjectures#WeakDonovan conjecture|Weak Donovan conjecture]] for blocks of quasisimple groups with abelian defect groups. We may further assume that the centre of the group is a <math>p'</math>-group. See [[References#E|[EL18b]]], [[References#F|[FK18]]].<br />
<br />
<br />
=== <math>\mathcal{O}</math>-Donovan conjecture ===<br />
<br />
Eisele in [[References#E|[Ei18]]] proved the analogue of [[References#K|[Kü95]]] for the <math>\mathcal{O}</math>-Donovan conjecture, so it suffices to consider blocks of finite groups that are generated by the defect groups.<br />
<br />
By [[References#E|[EL20]]] in order to verify the <math>\mathcal{O}</math>-Donovan conjecture for a <math>p</math>-group <math>P</math> it suffices to check it for blocks of finite groups <math>G</math> with defect group <math>D \cong P</math> and no proper normal subgroup <math>N \triangleleft G</math> such that <math>G=C_D(D \cap N)N</math>. <br />
<br />
'''<math>P</math> abelian:''' To show the <math>\mathcal{O}</math>-Donovan conjecture for abelian <math>p</math>-groups, it suffices to verify the [[Statements of conjectures#WeakDonovan conjecture|Weak Donovan conjecture]] for blocks of quasisimple groups with abelian defect groups. We may further assume that the centre of the group is a <math>p'</math>-group. See [[References#E|[EEL18]]], [[References#F|[FK18]]].<br />
<br />
'''<math>3_+^{1+2}</math>:''' To show the <math>\mathcal{O}</math>-Donovan conjecture for <math>3_+^{1+2}</math>, it suffices to verify the [[Statements of conjectures#WeakDonovan conjecture|Weak Donovan conjecture]] for blocks of quasisimple groups with defect groups <math>3_+^{1+2}</math>. See [[References#A|[AE23]]].<br />
<br />
== Weak Donovan conjecture ==<br />
<br />
For arbitrary <math>p</math>-groups, it suffices to check the conjecture for blocks of quasisimple groups with centre of order not divisible by <math>p</math>. See [[References#D|[Du04]]].</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Reductions&diff=1212
Reductions
2023-10-03T08:33:20Z
<p>Charles Eaton: Extraspecial</p>
<hr />
<div>This page will contain descriptions of reduction techniques and results.<br />
<br />
== Donovan's conjecture ==<br />
<br />
[[Statements of conjectures#Donovan's conjecture|For the statement of the conjecture click here.]]<br />
<br />
=== <math>k</math>-Donovan conjecture ===<br />
<br />
By [[References#K|[Kü95]]] it suffices to consider blocks of finite groups that are generated by the defect groups, i.e., the defect groups are contained in no proper normal subgroup.<br />
<br />
Several reductions were achieved in [[References#D|[Du04]]], but these have been subsumed in later work.<br />
<br />
'''<math>P</math> abelian:''' To show the <math>k</math>-Donovan conjecture for abelian <math>p</math>-groups, it suffices to verify the [[Statements of conjectures#WeakDonovan conjecture|Weak Donovan conjecture]] for blocks of quasisimple groups with abelian defect groups. We may further assume that the centre of the group is a <math>p'</math>-group. See [[References#E|[EL18b]]], [[References#F|[FK18]]].<br />
<br />
<br />
=== <math>\mathcal{O}</math>-Donovan conjecture ===<br />
<br />
Eisele in [[References#E|[Ei18]]] proved the analogue of [[References#K|[Kü95]]] for the <math>\mathcal{O}</math>-Donovan conjecture, so it suffices to consider blocks of finite groups that are generated by the defect groups.<br />
<br />
By [[References#E|[EL20]]] in order to verify the <math>\mathcal{O}</math>-Donovan conjecture for a <math>p</math>-group <math>P</math> it suffices to check it for blocks of finite groups <math>G</math> with defect group <math>D \cong P</math> and no proper normal subgroup <math>N \triangleleft G</math> such that <math>G=C_D(D \cap N)N</math>. <br />
<br />
'''<math>P</math> abelian:''' To show the <math>\mathcal{O}</math>-Donovan conjecture for abelian <math>p</math>-groups, it suffices to verify the [[Statements of conjectures#WeakDonovan conjecture|Weak Donovan conjecture]] for blocks of quasisimple groups with abelian defect groups. We may further assume that the centre of the group is a <math>p'</math>-group. See [[References#E|[EEL18]]], [[References#F|[FK18]]].<br />
<br />
'''<math>3_+^{1+2}</math>:''' To show the <math>\mathcal{O}</math>-Donovan conjecture for <math>3_+^{1+2}</math>, it suffices to verify the [[Statements of conjectures#WeakDonovan conjecture|Weak Donovan conjecture]] for blocks of quasisimple groups with defect groups <math>3_+^{1+2}</math>.<br />
<br />
== Weak Donovan conjecture ==<br />
<br />
For arbitrary <math>p</math>-groups, it suffices to check the conjecture for blocks of quasisimple groups with centre of order not divisible by <math>p</math>. See [[References#D|[Du04]]].</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=References&diff=1211
References
2023-10-03T08:30:27Z
<p>Charles Eaton: Added [AE23]</p>
<hr />
<div>[[#A|A,]] [[#B|B,]] [[#C|C,]] [[#D|D,]] [[#E|E,]] [[#F|F,]] [[#G|G,]] [[#H|H,]] [[#I|I,]] [[#J|J,]] [[#K|K,]] [[#L|L,]] [[#M|M,]] [[#N|N,]] [[#O|O,]] [[#P|P,]] [[#Q|Q,]] [[#R|R,]] [[#S|S,]] [[#T|T]] [[#U|U,]] [[#V|V,]] [[#W|W,]] [[#X|X,]] [[#Y|Y,]] [[#Z|Z,]] <br />
<br />
{| <br />
|- id="A"<br />
|[Al79] || '''J. L. Alperin''', ''Projective modules for <math>SL(2,2^n)</math>'', J. Pure and Applied Algebra '''15''' (1979), 219-234.<br />
|-<br />
|[Al80] || '''J. L. Alperin''', ''Local representation theory'', The Santa Cruz Conference on Finite Groups., Proc. Sympos. Pure Math. '''37''' (1980), 369-375.<br />
|-<br />
|[AE81] || '''J. L. Alperin and L. Evens''', ''Representations, resoluutions and Quillen's dimension theorem'', J. Pure Appl. Algebra '''22''' (1981), 1-9.<br />
|-<br />
|[AE04] || '''Jianbei An and C. W. Eaton''', ''Blocks with trivial intersection defect groups'', Math. Z. '''247''' (2004), 461-486.<br />
|-<br />
|[AE04] || '''Jianbei An and C. W. Eaton''', ''Morita equivalence classes of blocks with extraspecial defect groups <math>p_+^{1+2}</math>''<br />
|-<br />
|[Ar19] || '''C. G. Ardito''', [https://arxiv.org/abs/1908.02652 ''Morita equivalence classes of blocks with elementary abelian defect groups of order 32''], J. Algebra '''573''' (2021), 297-335.<br />
|-<br />
|[ArMcK20] || '''C. G. Ardito and E. McKernon''', ''[https://arxiv.org/abs/2010.08329 ''2-blocks with an abelian defect group and a freely acting cyclic inertial quotient''], [https://arxiv.org/abs/2010.08329 arxiv.org/abs/2010.08329]<br />
|-<br />
|[AS20] || '''C. G. Ardito and B. Sambale''', [http://www.advgrouptheory.com/journal/Volumes/12/ArditoSambale.pdf ''Broué's Conjecture for 2-blocks with elementary abelian defect groups of order 32''], Advances in Group Theory and Applications 12 (2021), 71–78. <br />
|-<br />
|[AKO11] || '''M. Aschbacher, R. Kessar and B. Oliver''', ''Fusion systems in algebra and topology'', London Mathematical Society Lecture Notes '''391''', Cambridge University Press (2011).<br />
|- id="B"<br />
|[BK07] || '''D. Benson and R. Kessar''', ''Blocks inequivalent to their Frobenius twists'', J. Algebra '''315''' (2007), 588-599.<br />
|-<br />
|[BKL18] || '''R. Boltje, R. Kessar, and M. Linckelmann''', [https://doi.org/10.1016/j.jalgebra.2019.02.045 ''On Picard groups of blocks of finite groups''], J. Algebra '''558''' (2020), 70-101.<br />
|-<br />
|[Bra41] || '''R. Brauer''', ''Investigations on group characters'', Ann. Math. '''42''' (1941), 936-958.<br />
|-<br />
|[BP80] || '''M. Broué and L. Puig''', ''A Frobenius theorem for blocks'', Invent. Math. '''56''' (1980), 117-128.<br />
|-<br />
|[BP80b] || '''M. Broué and L. Puig''', ''Characters and local structure in G-algebras'', J. Algebra '''63''' (1980), 306-317.<br />
|- id="C"<br />
|[Cr11] || '''D. A. Craven''', ''The Theory of Fusion Systems: An Algebraic Approach'', Cambridge University Press (2011).<br />
|-<br />
|[Cr12] || '''D. A. Craven''', [https://arxiv.org/abs/1207.0116 ''Perverse Equivalences and Broué's Conjecture II: The Cyclic Case''], [https://arxiv.org/abs/1207.0116 arXiv:1207.0116]<br />
|-<br />
|[CDR18] || '''D. A. Craven, O. Dudas and R. Rouquier''', [https://arxiv.org/abs/1701.07097 ''The Brauer trees of unipotent blocks''], to appear, J. EMS, [https://arxiv.org/abs/1701.07097 arXiv:1701.07097] <br />
|-<br />
|[CEKL11] || '''D. A. Craven, C. W. Eaton, R. Kessar and M. Linckelmann''', ''The structure of blocks with a Klein four defect group'', Math. Z. '''268''' (2011), 441-476.<br />
|-<br />
|[CG12] || '''D. A. Craven and A. Glesser''', ''Fusion systems on small p-groups'', Trans. AMS '''364''' (2012) 5945-5967.<br />
|-<br />
|[CR13] || '''D. A. Craven and R. Rouquier''', ''Perverse equivalences and Broué's conjecture'', Adv. Math. '''248''' (2013), 1-58.<br />
|-<br />
|[CuRe81a] || '''C. W. Curtis and I. Reiner''', ''Methods of representation theory, with applications to finite groups and orders, Volume I'', Wiley-Interscience (1981).<br />
|-<br />
|[CuRe81b] || '''C. W. Curtis and I. Reiner''', ''Methods of representation theory, with applications to finite groups and orders, Volume II'', Wiley-Interscience (1981).<br />
|- id="D"<br />
|[Da66] || '''E. C. Dade''', ''Blocks with cyclic defect groups'', Ann. Math. '''84''' (1966), 20-48. <br />
|-<br />
|[DE20] || '''S. Danz and K. Erdmann''', [https://arxiv.org/abs/2008.10999 ''On Ext-Quivers of Blocks of weight two for symmetric groups''], [https://arxiv.org/abs/2008.10999 arXiv:2008.10999]<br />
|-<br />
|[Du14] || '''O. Dudas''', [https://arxiv.org/abs/1011.5478 ''Coxeter orbits and Brauer trees II''], Int. Math. Res. Not. '''15''' (2014), 4100-4123.<br />
|-<br />
|[Dü04] || '''O. Düvel''', ''On Donovan's conjecture'', J. Algebra '''272''' (2004), 1-26.<br />
|- id="E"<br />
|[Ea16] || '''C. W. Eaton''', ''Morita equivalence classes of <math>2</math>-blocks of defect three'', Proc. AMS '''144''' (2016), 1961-1970.<br />
|-<br />
|[Ea18] || '''C. W. Eaton''', [https://arxiv.org/abs/1612.03485 ''Morita equivalence classes of blocks with elementary abelian defect groups of order 16''], [https://arxiv.org/abs/1612.03485 arXiv:1612.03485]<br />
|-<br />
|[EEL18] || '''C. W. Eaton, F. Eisele and M. Livesey''', [https://arxiv.org/abs/1809.08152 ''Donovan's conjecture, blocks with abelian defect groups and discrete valuation rings''], Math. Z. '''295''' (2020), 249-264.<br />
|-<br />
|[EKKS14] || '''C. W. Eaton, R. Kessar, B. Külshammer and B. Sambale''', ''<math>2</math>-blocks with abelian defect groups'', Adv. Math. '''254''' (2014), 706-735.<br />
|-<br />
|[EKS12] || '''C. W. Eaton, B. Külshammer and B. Sambale''', ''<math>2</math>-blocks with minimal nonabelian defect groups, II'', J. Group Theory '''15''' (2012), 311-321.<br />
|-<br />
|[EL18a] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/1709.04331 Classifying blocks with abelian defect groups of rank 3 for the prime 2]'', J. Algebra '''515''' (2018), 1-18.<br />
|-<br />
|[EL18b] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/1803.03539 Donovan's conjecture and blocks with abelian defect groups]'', Proc. AMS. '''147''' (2019), 963-970.<br />
|-<br />
|[EL18c] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/1810.10950 Some examples of Picard groups of blocks]'', J. Algebra '''558''' (2020), 350-370.<br />
|-<br />
|[EL20] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/2006.11173 Donovan's conjecture and extensions by the centralizer of a defect group]'', J. Algebra '''582''' (2021), 157-176.<br />
|-<br />
|[Ei16] || '''F. Eisele''', ''Blocks with a generalized quaternion defect group and three simple modules over a <math>2</math>-adic ring'', J. Algebra '''456''' (2016), 294-322.<br />
|-<br />
|[Ei18] || '''F. Eisele''', ''[https://arxiv.org/abs/1807.05110 The Picard group of an order and Külshammer reduction]'', Algebr. Represent. Theory '''24''' (2021), 505-518.<br />
|-<br />
|[Ei19] || '''F. Eisele''', ''[https://arxiv.org/abs/1908.00129 On the geometry of lattices and finiteness of Picard groups]'', [https://arxiv.org/abs/1908.00129 arXiv:1908.00129]<br />
|-<br />
|[EiLiv20] || '''F. Eisele and M. Livesey''', ''[https://arxiv.org/abs/2006.13837 Arbitrarily large Morita Frobenius numbers]'', [https://arxiv.org/abs/2006.13837 arXiv:2006.13837]<br />
|-<br />
|[Er82] || '''K. Erdmann''', ''Blocks whose defect groups are Klein four groups: a correction'', J. Algebra '''76''' (1982), 505-518.<br />
|-<br />
|[Er87] || '''K. Erdmann''', ''Algebras and dihedral defect groups'', Proc. LMS '''54''' (1987), 88-114.<br />
|-<br />
|[Er88a] || '''K. Erdmann''', ''Algebras and quaternion defect groups, I'', Math. Ann. '''281''' (1988), 545-560.<br />
|-<br />
|[Er88b] || '''K. Erdmann''', ''Algebras and quaternion defect groups, II'', Math. Ann. '''281''' (1988), 561-582. <br />
|-<br />
|[Er88c] || '''K. Erdmann''', ''Algebras and semidihedral defect groups I'', Proc. LMS '''57''' (1988), 109-150. <br />
|-<br />
|[Er90] || ''' K. Erdmann''', ''Blocks of tame representation type and related algebras'', Lecture Notes in Mathematics '''1428''', Springer-Verlag (1990).<br />
|-<br />
|[Er90b] || '''K. Erdmann''', ''Algebras and semidihedral defect groups II'', Proc. LMS '''60''' (1990), 123-165.<br />
|- id="F"<br />
|[Fa17] || '''N. Farrell''', ''On the Morita Frobenius numbers of blocks of finite reductive groups'', J. Algebra '''471''' (2017), 299-318.<br />
|-<br />
|[FK18] || '''N. Farrell and R. Kessar''', [https://arxiv.org/abs/1805.02015 ''Rationality of blocks of quasi-simple finite groups''], Represent. Theory '''23''' (2019), 325-349.<br />
|- id="G"<br />
|[GMdelR21] || '''D. Garcia, l. Margolis and A. del Rio''', [https://arxiv.org/abs/2016.07231 ''Non-isomorphic 2-groups with isomorphic modular group algebras''], J. Reine Angew. Math. '''f783''' (2022), 269–274.<br />
|-<br />
|[GO97] || '''H. Gollan and T. Okuyama''', ''Derived equivalences for the smallest Janko group'', preprint (1997).<br />
|-<br />
|[GT19] || '''R. M. Guralnick and Pham Huu Tiep''', ''Sectional rank and Cohomology'', J. Algebra (2019) https://doi.org/10.1016/j.jalgebra.2019.04.023<br />
|- id="H"<br />
|[HM07] || '''G. T. Helleloid and U. Martin''', ''The automorphism group of a finite <math>p</math>-group is almost always a <math>p</math>-group'', J. Algebra (2007), 294-329.<br />
|-<br />
|[HP94] || '''H-W. Henn and S. Priddy''', ''<math>p</math>-nilpotence, classifying space indecompsability, and other properties of almost finite groups'', Comment. Math. Helvetici (1994), 335-350.<br />
|- <br />
|[HK00] || '''G. Hiss and R. Kessar''', ''Scopes reduction and Morita equivalence classes of blocks in finite classical groups'', J. Algebra '''230''' (2000), 378-423.<br />
|-<br />
|[HK05] || '''G. Hiss and R. Kessar''', ''Scopes reduction and Morita equivalence classes of blocks in finite classical groups II'', J. Algebra '''283''' (2005), 522-563.<br />
|-<br />
|[Ho97] || '''T. Holm''', ''Derived equivalent tame blocks'', J. Algebra '''194''' (1997), 178-200.<br />
|-<br />
|[HKL07] || '''T. Holm, R. Kessar and M. Linckelmann''', ''Blocks with a quaternion defect group over a 2-adic ring: the case <math>\tilde{A}_4</math>'', Glasgow Math. J. '''49''' (2007), 29–43.<br />
|- id="J"<br />
|[Ja69] || '''G. Janusz''', ''Indecomposable modules for finite groups'', Ann. Math. '''89''' (1969), 209-241.<br />
|-<br />
|[Jo96] || '''T. Jost''', ''Morita equivalences for blocks of finite general linear groups'', Manuscripta Math. '''91''' (1996), 121-144.<br />
|- id="K"<br />
|[Ke96] || '''R. Kessar''', ''Blocks and source algebras for the double covers of the symmetric and alternating groups'', J. Algebra '''186''' (1996), 872-933.<br />
|-<br />
|[Ke00] || '''R. Kessar''', ''Equivalences for blocks of the Weyl groups'', Proc. Amer. Math. Soc. '''128''' (2000), 337-346.<br />
|-<br />
|[Ke01] || '''R. Kessar''', ''Source algebra equivalences for blocks of finite general linear groups over a fixed field'', Manuscripta Math. '''104''' (2001), 145-162. <br />
|-<br />
|[Ke02] || '''R. Kessar''', ''Scopes reduction for blocks of finite alternating groups'', Quart. J. Math. '''53''' (2002), 443-454.<br />
|-<br />
|[Ke05] || ''' R. Kessar''', ''A remark on Donovan's conjecture'', Arch. Math (Basel) '''82''' (2005), 391-394.<br />
|-<br />
|[KL18] || '''R. Kessar and M. Linckelmann''', [https://arxiv.org/abs/1705.07227 ''Descent of equivalences and character bijections''], [https://arxiv.org/abs/1705.07227 arXiv:1705.07227]<br />
|-<br />
|[Ki84] || '''M. Kiyota''', ''On 3-blocks with an elementary abelian defect group of order 9'', J. Fac. Sci. Univ. Tokyo Sect. IA Math. '''31''' (1984), 33–58.<br />
|-<br />
|[Ko03] || '''S. Koshitani''', ''Conjectures of Donovan and Puig for principal <math>3</math>-blocks with abelian defect groups'', Comm. Alg. '''31''' (2003), 2229-2243; ''Corrigendum'', '''32''' (2004), 391-393.<br />
|-<br />
|[KKW02] || '''S. Koshitani, N. Kunugi and K. Waki''', ''Broué's conjecture for non-principal 3-blocks of finite groups'', J. Pure and Applied Algebra '''173''' (2002), 177-211. <br />
|-<br />
|[KKW04] || '''S. Koshitani, N. Kunugi and K. Waki''', ''Broué's abelian defect group conjecture for Held group and the sporadic Suzuki group'', J. Algebra '''279''' (2004), 638-666. <br />
|-<br />
|[KoLa20] || '''S. Koshitani and C. Lassueur''', ''Splendid Morita equivalences for principal 2-blocks with dihedral defect groups'', Math. Z. '''294''' (2020), 639-666.<br />
|-<br />
|[KoLa20b] || '''S. Koshitani and C. Lassueur''', ''Splendid Morita equivalences for principal blocks with generalised quaternion defect groups'', J. Algebra '''558''' (2020), 523-533.<br />
|-<br />
|[KoLaSa22] || '''S. Koshitani, C. Lassueur and B. Sambale''', ''Splendid Morita equivalences for principal blocks with semidihedral defect groups'', Proceedings of the American Mathematical Society '''150''' (2022), 41-53.<br />
|-<br />
|[Kü80] || '''B. Külshammer''', ''On 2-blocks with wreathed defect groups'', J. Algebra '''64''' (1980), 529–555.<br />
|-<br />
|[Kü81] || '''B. Külshammer''', ''On p-blocks of p-solvable groups'', Comm. Alg. '''9''' (1981), 1763-1785.<br />
|-<br />
|[Kü91] || '''B. Külshammer''', ''Group-theoretical descriptions of ring-theoretical invariants of group algebras'', in Representation Theory of Finite Groups and Finite-Dimensional Algebras (Bielefeld, 1991), Progr. Math. '''95''', pp. 425-442, Birkhauser (1991).<br />
|-<br />
|[Kü95] || '''B. Külshammer''', ''Donovan's conjecture, crossed products and algebraic group actions'', Israel J. Math. '''92''' (1995), 295-306.<br />
|-<br />
|[KS13] || '''B. Külshammer and B. Sambale''', ''The 2-blocks of defect 4'', Representation Theory '''17''' (2013), 226-236.<br />
|-<br />
|[Ku00] || '''N. Kunugi''', ''Morita equivalent 3-blocks of the 3-dimensional projective special linear groups'', Proc. LMS '''80''' (2000), 575-589.<br />
|-<br />
|[Kup69] || '''H. Kupisch''', ''Unzerlegbare Moduln endlicher Gruppen mit zyklischer p-Sylow Gruppe'', Math. Z. '''108''' (1969), 77-104.<br />
|- id="L"<br />
|[LM80]||'''P. Landrock and G. O. Michler''', ''Principal 2-blocks of the simple groups of Ree type'', Trans. AMS '''260''' (1980), 83-111.<br />
|-<br />
|[Li94] || '''M. Linckelmann''', ''The source algebras of blocks with a Klein four defect group'', J. Algebra '''167''' (1994), 821-854.<br />
|-<br />
|[Li94b] || '''M. Linckelmann''', ''A derived equivalence for blocks with dihedral defect groups'', J. Algebra '''164''' (1994), 244-255. <br />
|-<br />
|[Li96] || '''M. Linckelmann''', ''The isomorphism problem for cyclic blocks and their source algebras'', Invent. Math. '''125''' (1996), 265-283.<br />
|-<br />
|[Li18] || '''M. Linckelmann''', [https://arxiv.org/abs/1805.08884 ''The strong Frobenius numbers for cyclic defect blocks are equal to one''], [https://arxiv.org/abs/1805.08884 arXiv:1805.08884]<br />
|-<br />
|[Li18b] || '''M. Linckelmann''', ''Finite-dimensional algebras arising as blocks of finite group algebras'', Contemporary Mathematics '''705''' (2018), 155-188.<br />
|-<br />
|[Li18c] || '''M. Linckelmann''', ''The block theory of finite group algebras, Volume 1'', London Math. Soc. Student Texts '''92''', Cambridge University Press (2018).<br />
|-<br />
|[Li18d] || '''M. Linckelmann''', ''The block theory of finite group algebras, Volume 2'', London Math. Soc. Student Texts '''92''', Cambridge University Press (2018).<br />
|-<br />
|[LM20] || '''M. Linckelmann and W. Murphy''', [https://arxiv.org/abs/2005.02223 ''A 9-dimensional algebra which is not a block of a finite group''], Quarterly Journal of Mathematics 72 (2021), 1077–1088<br />
|-<br />
|[Liv19] || '''M. Livesey''', [https://arxiv.org/abs/1907.12167 ''On Picard groups of blocks with normal defect groups''], J. Algebra '''566''' (2021), 94-118.<br />
|-<br />
|[LiMa20] || '''M. Livesey and C. Marchi''', [https://arxiv.org/abs/2002.10571 ''On Picent for blocks with normal defect group''], [https://arxiv.org/abs/2002.10571 arXiv:2002.10571]<br />
|-<br />
|[LiMa20b] || '''M. Livesey and C. Marchi''', [https://arxiv.org/abs/2008.05857 ''Picard groups for blocks with normal defect groups and linear source bimodules''], [https://arxiv.org/abs/2008.05857 arXiv:2008.05857]<br />
|- id="M"<br />
|[Mac] || '''N. Macgregor''', ''Morita equivalence classes of tame blocks of finite groups'', J. Algebra '''608''' (2022), 719-754.<br />
|-<br />
|[Mar] || '''C. Marchi''', ''Picard groups for blocks'', PhD thesis, University of Manchester (2022)<br />
|-<br />
|[Ma86] || '''U. Martin''', ''Almost all <math>p</math>-groups have automorphism group a <math>p</math>-group'', Bull. AMS '''15''' (1986), 78-82.<br />
|-<br />
|[McK19] || '''E. McKernon''', [https://arxiv.org/abs/1912.03222 ''2-Blocks whose defect group is homocyclic and whose inertial quotient contains a Singer cycle''], J. Algebra '''563''' (2020), 30–48.<br />
|-<br />
|[MS08] || '''J. Müller and M. Schaps''', ''The Broué conjecture for the faithful 3-blocks of <math>4.M_{22}</math>'', J. Algebra '''319''' (2008), 3588-3602.<br />
|- id="N"<br />
|[NS18] || '''G. Navarro and B. Sambale''', ''On the blockwise modular isomorphism problem'', Manuscripta Math. '''157''' (2018), 263-278.<br />
|- <br />
|[Ne02] || '''G. Nebe''', [http://www.math.rwth-aachen.de/~Gabriele.Nebe/papers/survey.pdf ''Group rings of finite groups over p-adic integers, some examples''], Proceedings of the conference Around Group rings (Edmonton) Resenhas '''5''' (2002), 329-350.<br />
|- id="O"<br />
|[Ok97] || '''T. Okuyama''', ''Some examples of derived equivalent blocks of finite groups'', preprint (1997).<br />
|- id="P"<br />
|[Pu88]|| '''L. Puig''', ''Nilpotent blocks and their source algebras'', Invent. Math. '''93''' (1988), 77-116.<br />
|-<br />
|[Pu94] || '''L. Puig''', ''On Joanna Scopes’ criterion of equivalence for blocks of symmetric groups'', Algebra Colloq. '''1''' (1994), 25-55.<br />
|-<br />
|[Pu99]|| '''L. Puig''', ''On the local structure of Morita and Rickard equivalences between Brauer blocks'', Progress in Math. '''178''', Birkhauser Verlag (1999).<br />
|-<br />
|[Pu09] || '''L. Puig''', ''Block source algebras in p-solvable groups'', Michigan Math. J. '''58''' (2009), 323-338.<br />
|- id="R"<br />
|[Ri96] || '''J. Rickard''', ''Splendid equivalences: derived categories and permutation modules'', Proc. London Math. Soc. '''72''' (1996), 331-358.<br />
|-<br />
|[Ro95] || '''R. Rouquier''', ''From stable equivalences to Rickard equivalences for blocks with cyclic defect'', Proceedings of Groups 1993, Galway-St. Andrews Conference, Vol. 2, London Math. Soc. Lecture Note Ser. '''212''', Cambridge University Press (1995), 512-523.<br />
|-<br />
|[Ru11] || '''P. Ruengrot''', ''Perfect isometry groups for blocks of finite groups'', PhD Thesis, University of Manchester (2011).<br />
|- id="S"<br />
|[Sa11] || '''B. Sambale''', ''<math>2</math>-blocks with minimal nonabelian defect groups'', J. Algebra '''337''' (2011), 261–284.<br />
|-<br />
|[Sa12] || '''B. Sambale''', ''Blocks with defect group <math>D_{2^n} \times C_{2^m}</math>'', J. Pure Appl. Algebra '''216''' (2012), 119–125.<br />
|-<br />
|[Sa12b] || '''B. Sambale''', ''Fusion systems on metacyclic 2-groups'', Osaka J. Math. '''49''' (2012), 325–329.<br />
|-<br />
|[Sa13] || '''B. Sambale''', ''Blocks with defect group <math>Q_{2^n} \times C_{2^m}</math> and <math>SD_{2^n} \times C_{2^m}</math>'', Algebr. Represent. Theory '''16''' (2013), 1717–1732.<br />
|-<br />
|[Sa13b] || '''B. Sambale''', ''Blocks with central product defect group <math>D_{2^n} ∗ C_{2^m}</math>'', Proc. Amer. Math. Soc. '''141''' (2013), 4057–4069.<br />
|-<br />
|[Sa13c] || '''B. Sambale''', ''Further evidence for conjectures in block theory'', Algebra Number Theory '''7''' (2013), 2241–2273. <br />
|-<br />
|[Sa14] || '''B. Sambale''', ''Blocks of Finite Groups and Their Invariants'', Lecture Notes in Mathematics, Springer (2014).<br />
|-<br />
|[Sa16] || '''B. Sambale''', ''2-blocks with minimal nonabelian defect groups III'', Pacific J. Math. '''280''' (2016), 475–487.<br />
|-<br />
|[Sa20] || '''B. Sambale''', [https://arxiv.org/abs/2005.13172 ''Blocks with small-dimensional basic algebra''], Bul. Aust. Math. Soc. '''103''' (2021), 461-474.<br />
|-<br />
|[SSS98] || '''M. Schaps, D. Shapira and O. Shlomo''', ''Quivers of blocks with normal defect groups'', Proc. Symp. in Pure Mathematics '''63''', Amer. Math. Soc. (1998), 497-510.<br />
|-<br />
|[Sc91] || '''J. Scopes''', ''Cartan matrices and Morita equivalence for blocks of the symmetric groups'', J. Algebra '''142''' (1991), 441-455.<br />
|-<br />
|[Sh20] || '''V. Shalotenko''', ''Bounds on the dimension of Ext for finite groups of Lie type'', J. Algebra '''550''' (2020), 266-289.<br />
|-<br />
|[St02] || '''R. Stancu''', ''Almost all generalized extraspecial p-groups are resistant'', J. Algebra '''249''' (2002), 120-126.<br />
|-<br />
|[St06] || '''R. Stancu''', ''Control of fusion in fusion systems'', J. Algebra and its Applications '''5''' (2006), 817-837. <br />
|- id="T"<br />
|[Th93] || '''J. Thévenaz''', ''Most finite groups are <math>p</math>-nilpotent'', Exposition. Math. '''11''' (1993), 359-363.<br />
|- id="V"<br />
|[vdW91] || '''R. van der Waall''', ''On p-nilpotent forcing groups'', Indag. Mathem., N.S., '''2''' (1991), 367-384.<br />
|- id="W"<br />
|[Wa00]|| '''A. Watanabe''', ''A remark on a splitting theorem for blocks with abelian defect groups'', RIMS Kokyuroku '''1140''', Edited by H.Sasaki, Research Institute for Mathematical Sciences, Kyoto University (2000), 76-79.<br />
|-<br />
|[WZZ18] || '''Chao Wu, Kun Zhang and Yuanyang Zhou''', ''[https://arxiv.org/abs/1803.07262 Blocks with defect group <math>Z_{2^n} \times Z_{2^n} \times Z_{2^m}</math>]'', J. Algebra '''510''' (2018), 469-498.<br />
|}</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Generic_classifications_by_p-group_class&diff=1210
Generic classifications by p-group class
2023-10-03T08:28:53Z
<p>Charles Eaton: Added extraspecial</p>
<hr />
<div>This page contains results for classes of ''p''-groups for which we either have classifications or have general results concerning Morita equivalence classes. <br />
<br />
== Fusion trivial ''p''-groups ==<br />
<br />
''p''-groups <math>P</math> for which the only saturated fusion system is <math>\mathcal{F}_P(P)</math> have not yet been given a name in the literature (to our knowledge). We will call them ''fusion trivial'', but ''nilpotent forcing'' also seems appropriate following [[References#V|[vdW91]]] (where ''p''-groups for which any finite group <math>G</math> containing <math>P</math> as a Sylow ''p''-subgroup must be ''p''-nilpotent are called ''p-nilpotent forcing''). It is not known whether these two definitions are equivalent, i.e., whether there exist ''p''-nilpotent forcing ''p''-groups for which there is an exotic fusion system.<br />
<br />
Blocks with fusion trivial defect groups must be nilpotent and so Morita equivalent to the group algebra of a defect group by [[References#P|[Pu88]]].<br />
<br />
Examples of fusion trivial ''p''-groups are abelian ''2''-groups with automorphism group a ''2''-group (i.e., those whose cyclic factors have pairwise distinct orders), and metacyclic 2-groups other than homocyclic, dihedral, generalised quaternion or semidihedral groups (see [[References#C|[CG12]]] or [[References#S|[Sa12b]]]).<br />
<br />
Note that a ''p''-group is fusion trivial if and only if it is resistant and has automorphism group a ''p''-group. See [[References#S|[St06]]] for an analysis of resistant ''p''-groups.<br />
<br />
== Cyclic ''p''-groups ==<br />
<br />
[[Blocks with cyclic defect groups|Click here for background on blocks with cyclic defect groups]].<br />
<br />
Morita equivalence classes are labelled by [[Brauer trees]], but it is at present an open problem as to which Brauer trees are realised by blocks of finite groups. Each ''k''-Morita equivalence class corresponds to an unique <math>\mathcal{O}</math>-Morita equivalence class. <br />
<br />
For <math>p=2,3</math> every appropriate Brauer tree is realised by a block and we can give generic descriptions.<br />
<br />
[[C(2^n)|<math>2</math>-blocks with cyclic defect groups]]<br />
<br />
[[C(3^n)|<math>3</math>-blocks with cyclic defect groups]]<br />
<br />
== Tame blocks ==<br />
<br />
[[Tame blocks|Click here for background on tame blocks]].<br />
<br />
Erdmann classified algebras which are candidates for [[Basic algebras|basic algebras]] of tame blocks, i.e., those with dihedral, semidihedral or generalised quaternion defect groups (see [[References|[Er90] ]]) and in the cases of dihedral and semihedral defect groups determined which are realised by blocks of finite groups. In the case of generalised quaternion groups, the case of blocks with two simple modules is still open. These classifications only hold with respect to the field ''k'' at present.<br />
<br />
Principal blocks with dihedral defect groups are classified up to source algebra equivalence in [[References#K|[KoLa20]]].<br />
<br />
== Abelian <math>2</math>-groups with <math>2</math>-rank at most three ==<br />
<br />
These have been classified in [[References#W|[WZZ18]]] and [[References#E|[EL18a]]] with respect to <math>\mathcal{O}</math>. The derived equivalences classes with respect to <math>\mathcal{O}</math> are known.<br />
<br />
Let <math>l,m,n \geq 1</math> be distinct with <math>l,m \neq 1</math><br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>rk_2(D) \leq 3</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>D</math> <br />
! scope="col"| # <math>(\mathcal{O}</math>-)Morita classes<br />
! scope="col"| # <math>\mathcal{O}</math>-Derived equiv classes<br />
! scope="col"| Always endopermutation source Morita equivalent?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|[[C(2^n)|<math>C_{2^n}</math>]] || 1 || 1 || Yes || [[References#L|[Li96]]] || [[Blocks with cyclic defect groups]]<br />
|-<br />
|[[C2xC2|<math>C_2 \times C_2</math>]] || 3 || 2 || Yes || [[References#L|[Li94]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)|<math>C_{2^m} \times C_{2^m}</math>]] || 2 || 2 || Yes || [[References#E|[EKKS14]]] ||<br />
|-<br />
|[[C(2^m)xC(2^n)|<math>C_{2^m} \times C_{2^n}</math>]] || 1 || 1 || Yes || [[References#P|[Pu88]]] || [[Nilpotent blocks]]<br />
|-<br />
|[[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]] || 8 || 4 || || [[References#E|[Ea16]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)xC(2^m)|<math>C_{2^m} \times C_{2^m} \times C_{2^m}</math>]] || 4 || 4 || Yes || [[References#W|[WZZ18]]] ||<br />
|-<br />
|[[C(2^m)xC2xC2|<math>C_{2^m} \times C_2 \times C_2</math>]] || 3 || 2 || || [[References#E|[EL18a]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)xC(2^n)|<math>C_{2^m} \times C_{2^m} \times C_{2^n}</math>]] || 2 || 2 || || [[References#W|[WZZ18]]] ||<br />
|-<br />
|[[C(2^l)xC(2^m)xC(2^n)|<math>C_{2^l} \times C_{2^m} \times C_{2^n}</math>]] || 1 || 1 || Yes || [[References#P|[Pu88]]] || [[Nilpotent blocks]]<br />
|}<br />
<br />
<!--== Abelian <math>2</math>-groups ==<br />
<br />
Donovan's conjecture holds for ''2''-blocks with abelian defect groups. Some generic classification results are known for certain inertial quotients. These will be detailed here.--><br />
<br />
== Minimal nonabelian <math>2</math>-groups ==<br />
<br />
Blocks with defect groups which are minimal nonabelian <math>2</math>-groups of the form <math>P=\langle x,y:x^{2^r}=y^{2^r}=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle</math> are classified in [[References#E|[EKS12]]]. There are two <math>\mathcal{O}</math>-Morita equivalence classes, with representatives <math>\mathcal{O}P</math> and <math>\mathcal{O}(P:C_3)</math>.<br />
<br />
For arbitrary minimal nonabelian <math>2</math>-groups, by [[References#S|[Sa16]]] blocks with such defect groups and the same fusion system are isotypic.<br />
<br />
== Homocyclic <math>2</math>-groups when inertial quotient contains a Singer cycle == <br />
<br />
A Singer cycle is an element of order <math>p^n-1</math> in <math>\operatorname{Aut}({C_p}^n) \cong GL_n(p)</math>, and a subgroup generated by a Singer cycle acts transitively on the non-trivial elements of <math>(C_p)^n</math>.<br />
<br />
<math>2</math>-blocks with homocyclic defect group <math> D \cong (C_{2^m})^n </math> whose inertial quotient <math> \mathbb{E} </math> contains a Singer cycle are classified in [[References#E|[McK19]]]. In this situation, <math>\mathbb{E}</math> has the form <math>E:F </math> where <math>E \cong C_{2^n-1}</math> and <math>F</math> is trivial or a subgroup of <math>C_n</math>. There are three <math> \mathcal{O} </math>-Morita equivalence classes when <math>m=1, n =3 </math>; two when <math>m=1 , n \neq 3 </math>; and only one when <math>m> 1 </math>. The three classes have representatives [[M(8,5,7) | <math> \mathcal{O}J_1 </math>]], which occurs only when <math>m=1, n=3</math>; <math> \mathcal{O}(SL_2(2^n):F) </math>, which occurs only when <math>m=1 </math>; and <math> \mathcal{O}(D : \mathbb{E})</math>. <br />
<br />
The Morita equivalence between the block and the class representative is known to be basic, possibly except when <math>m=1,n=3</math>, since the Morita equivalences between the [[M(8,5,8) |principal block of <math>{\rm \operatorname{Aut}(SL_2(8))}</math>]] and the blocks [[M(8,5,8)#Other_notatable_representatives |<math>B_0(\mathcal{O}({}^2G_2(3^{2m+1})))</math>]] are not known to be basic.<br />
<br />
<br />
== Abelian <math>2</math>-groups when inertial quotient is cyclic and acts freely on the defect group == <br />
<br />
<math>2</math>-blocks with abelian defect group <math> D </math> whose inertial quotient <math> E </math> is a cyclic group that acts freely on the defect group (i.e. such that the stabiliser in <math> E </math> of any nontrivial element of <math>D</math> is trivial) are classified in [[References#E|[ArMcK20]]].<br />
<br />
If the action of <math>E</math> is transitive, then <math>D</math> is homocyclic, <math>E</math> is a Singer cycle and, by the previous section, the block is either Morita equivalent to the principal block of <math> \mathcal{O}SL_2(2^n) </math>, or to <math> \mathcal{O}(D : E)</math>. <br />
<br />
If the action of <math>E</math> is not transitive, then the block is Morita equivalent to <math> \mathcal{O}(D : E)</math>. <br />
<br />
In each case, the Morita equivalence between the block and the class representative is known to be basic.<br />
<br />
== Extraspecial <math>p</math>-groups of order <math>p^3</math> and exponent <math>p</math> for <math>p \geq 5</math> ==<br />
<br />
These blocks are described in [[References#A|[AE23]]]. Donovan's conjecture holds in this case.</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=References&diff=1209
References
2023-07-28T14:21:28Z
<p>Charles Eaton: Updated [GMdelR21]</p>
<hr />
<div>[[#A|A,]] [[#B|B,]] [[#C|C,]] [[#D|D,]] [[#E|E,]] [[#F|F,]] [[#G|G,]] [[#H|H,]] [[#I|I,]] [[#J|J,]] [[#K|K,]] [[#L|L,]] [[#M|M,]] [[#N|N,]] [[#O|O,]] [[#P|P,]] [[#Q|Q,]] [[#R|R,]] [[#S|S,]] [[#T|T]] [[#U|U,]] [[#V|V,]] [[#W|W,]] [[#X|X,]] [[#Y|Y,]] [[#Z|Z,]] <br />
<br />
{| <br />
|- id="A"<br />
|[Al79] || '''J. L. Alperin''', ''Projective modules for <math>SL(2,2^n)</math>'', J. Pure and Applied Algebra '''15''' (1979), 219-234.<br />
|-<br />
|[Al80] || '''J. L. Alperin''', ''Local representation theory'', The Santa Cruz Conference on Finite Groups., Proc. Sympos. Pure Math. '''37''' (1980), 369-375.<br />
|-<br />
|[AE81] || '''J. L. Alperin and L. Evens''', ''Representations, resoluutions and Quillen's dimension theorem'', J. Pure Appl. Algebra '''22''' (1981), 1-9.<br />
|-<br />
|[AE04] || '''Jianbei An and C. W. Eaton''', ''Blocks with trivial intersection defect groups'', Math. Z. '''247''' (2004), 461-486.<br />
|-<br />
|[Ar19] || '''C. G. Ardito''', [https://arxiv.org/abs/1908.02652 ''Morita equivalence classes of blocks with elementary abelian defect groups of order 32''], J. Algebra '''573''' (2021), 297-335.<br />
|-<br />
|[ArMcK20] || '''C. G. Ardito and E. McKernon''', ''[https://arxiv.org/abs/2010.08329 ''2-blocks with an abelian defect group and a freely acting cyclic inertial quotient''], [https://arxiv.org/abs/2010.08329 arxiv.org/abs/2010.08329]<br />
|-<br />
|[AS20] || '''C. G. Ardito and B. Sambale''', [http://www.advgrouptheory.com/journal/Volumes/12/ArditoSambale.pdf ''Broué's Conjecture for 2-blocks with elementary abelian defect groups of order 32''], Advances in Group Theory and Applications 12 (2021), 71–78. <br />
|-<br />
|[AKO11] || '''M. Aschbacher, R. Kessar and B. Oliver''', ''Fusion systems in algebra and topology'', London Mathematical Society Lecture Notes '''391''', Cambridge University Press (2011).<br />
|- id="B"<br />
|[BK07] || '''D. Benson and R. Kessar''', ''Blocks inequivalent to their Frobenius twists'', J. Algebra '''315''' (2007), 588-599.<br />
|-<br />
|[BKL18] || '''R. Boltje, R. Kessar, and M. Linckelmann''', [https://doi.org/10.1016/j.jalgebra.2019.02.045 ''On Picard groups of blocks of finite groups''], J. Algebra '''558''' (2020), 70-101.<br />
|-<br />
|[Bra41] || '''R. Brauer''', ''Investigations on group characters'', Ann. Math. '''42''' (1941), 936-958.<br />
|-<br />
|[BP80] || '''M. Broué and L. Puig''', ''A Frobenius theorem for blocks'', Invent. Math. '''56''' (1980), 117-128.<br />
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|-<br />
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|-<br />
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|-<br />
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|-<br />
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|-<br />
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|-<br />
|[Ea18] || '''C. W. Eaton''', [https://arxiv.org/abs/1612.03485 ''Morita equivalence classes of blocks with elementary abelian defect groups of order 16''], [https://arxiv.org/abs/1612.03485 arXiv:1612.03485]<br />
|-<br />
|[EEL18] || '''C. W. Eaton, F. Eisele and M. Livesey''', [https://arxiv.org/abs/1809.08152 ''Donovan's conjecture, blocks with abelian defect groups and discrete valuation rings''], Math. Z. '''295''' (2020), 249-264.<br />
|-<br />
|[EKKS14] || '''C. W. Eaton, R. Kessar, B. Külshammer and B. Sambale''', ''<math>2</math>-blocks with abelian defect groups'', Adv. Math. '''254''' (2014), 706-735.<br />
|-<br />
|[EKS12] || '''C. W. Eaton, B. Külshammer and B. Sambale''', ''<math>2</math>-blocks with minimal nonabelian defect groups, II'', J. Group Theory '''15''' (2012), 311-321.<br />
|-<br />
|[EL18a] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/1709.04331 Classifying blocks with abelian defect groups of rank 3 for the prime 2]'', J. Algebra '''515''' (2018), 1-18.<br />
|-<br />
|[EL18b] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/1803.03539 Donovan's conjecture and blocks with abelian defect groups]'', Proc. AMS. '''147''' (2019), 963-970.<br />
|-<br />
|[EL18c] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/1810.10950 Some examples of Picard groups of blocks]'', J. Algebra '''558''' (2020), 350-370.<br />
|-<br />
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|-<br />
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|-<br />
|[EiLiv20] || '''F. Eisele and M. Livesey''', ''[https://arxiv.org/abs/2006.13837 Arbitrarily large Morita Frobenius numbers]'', [https://arxiv.org/abs/2006.13837 arXiv:2006.13837]<br />
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|-<br />
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|[LM80]||'''P. Landrock and G. O. Michler''', ''Principal 2-blocks of the simple groups of Ree type'', Trans. AMS '''260''' (1980), 83-111.<br />
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|[Li94] || '''M. Linckelmann''', ''The source algebras of blocks with a Klein four defect group'', J. Algebra '''167''' (1994), 821-854.<br />
|-<br />
|[Li94b] || '''M. Linckelmann''', ''A derived equivalence for blocks with dihedral defect groups'', J. Algebra '''164''' (1994), 244-255. <br />
|-<br />
|[Li96] || '''M. Linckelmann''', ''The isomorphism problem for cyclic blocks and their source algebras'', Invent. Math. '''125''' (1996), 265-283.<br />
|-<br />
|[Li18] || '''M. Linckelmann''', [https://arxiv.org/abs/1805.08884 ''The strong Frobenius numbers for cyclic defect blocks are equal to one''], [https://arxiv.org/abs/1805.08884 arXiv:1805.08884]<br />
|-<br />
|[Li18b] || '''M. Linckelmann''', ''Finite-dimensional algebras arising as blocks of finite group algebras'', Contemporary Mathematics '''705''' (2018), 155-188.<br />
|-<br />
|[Li18c] || '''M. Linckelmann''', ''The block theory of finite group algebras, Volume 1'', London Math. Soc. Student Texts '''92''', Cambridge University Press (2018).<br />
|-<br />
|[Li18d] || '''M. Linckelmann''', ''The block theory of finite group algebras, Volume 2'', London Math. Soc. Student Texts '''92''', Cambridge University Press (2018).<br />
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|[LM20] || '''M. Linckelmann and W. Murphy''', [https://arxiv.org/abs/2005.02223 ''A 9-dimensional algebra which is not a block of a finite group''], Quarterly Journal of Mathematics 72 (2021), 1077–1088<br />
|-<br />
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|-<br />
|[LiMa20] || '''M. Livesey and C. Marchi''', [https://arxiv.org/abs/2002.10571 ''On Picent for blocks with normal defect group''], [https://arxiv.org/abs/2002.10571 arXiv:2002.10571]<br />
|-<br />
|[LiMa20b] || '''M. Livesey and C. Marchi''', [https://arxiv.org/abs/2008.05857 ''Picard groups for blocks with normal defect groups and linear source bimodules''], [https://arxiv.org/abs/2008.05857 arXiv:2008.05857]<br />
|- id="M"<br />
|[Mac] || '''N. Macgregor''', ''Morita equivalence classes of tame blocks of finite groups'', J. Algebra '''608''' (2022), 719-754.<br />
|-<br />
|[Mar] || '''C. Marchi''', ''Picard groups for blocks'', PhD thesis, University of Manchester (2022)<br />
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|- id="N"<br />
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|- id="O"<br />
|[Ok97] || '''T. Okuyama''', ''Some examples of derived equivalent blocks of finite groups'', preprint (1997).<br />
|- id="P"<br />
|[Pu88]|| '''L. Puig''', ''Nilpotent blocks and their source algebras'', Invent. Math. '''93''' (1988), 77-116.<br />
|-<br />
|[Pu94] || '''L. Puig''', ''On Joanna Scopes’ criterion of equivalence for blocks of symmetric groups'', Algebra Colloq. '''1''' (1994), 25-55.<br />
|-<br />
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|-<br />
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|-<br />
|[Ru11] || '''P. Ruengrot''', ''Perfect isometry groups for blocks of finite groups'', PhD Thesis, University of Manchester (2011).<br />
|- id="S"<br />
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|-<br />
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|-<br />
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|-<br />
|[Sa13] || '''B. Sambale''', ''Blocks with defect group <math>Q_{2^n} \times C_{2^m}</math> and <math>SD_{2^n} \times C_{2^m}</math>'', Algebr. Represent. Theory '''16''' (2013), 1717–1732.<br />
|-<br />
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|-<br />
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|}</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=MNA(2,1)&diff=1202
MNA(2,1)
2022-11-09T11:47:28Z
<p>Charles Eaton: </p>
<hr />
<div>__NOTITLE__<br />
<br />
== Blocks with defect group <math>MNA(2,1)=\langle x,y|x^4=y^2=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle</math> ==<br />
<br />
The defect groups are minimal nonabelian <math>2</math>-groups. The invariants <math>k(B)</math>, <math>l(B)</math> and <math>k_i(B)</math> for all <math>i</math> are determined in [[References#S|[Sa11]]]. The Cartan matrices are also determined up to equivalence of quadratic forms. These results do not rely on the [[Glossary#CFSG|CFSG]]. The automorphism group of <math>MNA(2,1)</math> is a <math>2</math>-group, but by [[References#S|[Sa14,12.7]]] there exists precisely one non-nilpotent fusion system for blocks with this defect group, realised in SmallGroup(48,30) <math>\cong A_4:C_4</math>. By [[References#S|[Sa16]]] all non-nilpotent blocks with this defect group are [[Glossary#Isotypy|isotypic]]. <br />
'''<pre style="color: red">CLASSIFICATION NOT COMPLETE</pre>'''<br />
<br />
{| class="wikitable"<br />
|-<br />
! scope="col"| Class<br />
! scope="col"| Representative<br />
! scope="col"| # lifts / <math>\mathcal{O}</math><br />
! scope="col"| <math>k(B)</math><br />
! scope="col"| <math>l(B)</math><br />
! scope="col"| Inertial quotients<br />
! scope="col"| <math>{\rm Pic}_\mathcal{O}(B)</math><br />
! scope="col"| <math>{\rm Pic}_k(B)</math><br />
! scope="col"| <math>{\rm mf_\mathcal{O}(B)}</math><br />
! scope="col"| <math>{\rm mf_k(B)}</math><br />
! scope="col"| Notes<br />
<br />
|-<br />
|[[M(16,3,1)]] || <math>k(MNA(2,1))</math> || 1 ||10 ||1 ||<math>1</math> || || ||1 ||1 ||<br />
|-<br />
|[[M(16,3,2)]] || <math>B_0(k(A_5:C_4))</math> || ? ||10 ||2 ||<math>1</math> || || ||1 ||1 || <br />
|-<br />
|[[M(16,3,3)]] || <math>k(A_4:C_4)</math> || ? ||10 ||2 ||<math>1</math> || || ||1 ||1 || <br />
|}<br />
<br />
If <math>B</math> is not nilpotent, then <math>k(B)=10, k_1(B)=2, l(B)=2</math>.</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Generic_classifications_by_p-group_class&diff=1201
Generic classifications by p-group class
2022-09-07T13:08:10Z
<p>Charles Eaton: /* Fusion trivial p-groups */</p>
<hr />
<div>This page contains results for classes of ''p''-groups for which we either have classifications or have general results concerning Morita equivalence classes. <br />
<br />
== Fusion trivial ''p''-groups ==<br />
<br />
''p''-groups <math>P</math> for which the only saturated fusion system is <math>\mathcal{F}_P(P)</math> have not yet been given a name in the literature (to our knowledge). We will call them ''fusion trivial'', but ''nilpotent forcing'' also seems appropriate following [[References#V|[vdW91]]] (where ''p''-groups for which any finite group <math>G</math> containing <math>P</math> as a Sylow ''p''-subgroup must be ''p''-nilpotent are called ''p-nilpotent forcing''). It is not known whether these two definitions are equivalent, i.e., whether there exist ''p''-nilpotent forcing ''p''-groups for which there is an exotic fusion system.<br />
<br />
Blocks with fusion trivial defect groups must be nilpotent and so Morita equivalent to the group algebra of a defect group by [[References#P|[Pu88]]].<br />
<br />
Examples of fusion trivial ''p''-groups are abelian ''2''-groups with automorphism group a ''2''-group (i.e., those whose cyclic factors have pairwise distinct orders), and metacyclic 2-groups other than homocyclic, dihedral, generalised quaternion or semidihedral groups (see [[References#C|[CG12]]] or [[References#S|[Sa12b]]]).<br />
<br />
Note that a ''p''-group is fusion trivial if and only if it is resistant and has automorphism group a ''p''-group. See [[References#S|[St06]]] for an analysis of resistant ''p''-groups.<br />
<br />
== Cyclic ''p''-groups ==<br />
<br />
[[Blocks with cyclic defect groups|Click here for background on blocks with cyclic defect groups]].<br />
<br />
Morita equivalence classes are labelled by [[Brauer trees]], but it is at present an open problem as to which Brauer trees are realised by blocks of finite groups. Each ''k''-Morita equivalence class corresponds to an unique <math>\mathcal{O}</math>-Morita equivalence class. <br />
<br />
For <math>p=2,3</math> every appropriate Brauer tree is realised by a block and we can give generic descriptions.<br />
<br />
[[C(2^n)|<math>2</math>-blocks with cyclic defect groups]]<br />
<br />
[[C(3^n)|<math>3</math>-blocks with cyclic defect groups]]<br />
<br />
== Tame blocks ==<br />
<br />
[[Tame blocks|Click here for background on tame blocks]].<br />
<br />
Erdmann classified algebras which are candidates for [[Basic algebras|basic algebras]] of tame blocks, i.e., those with dihedral, semidihedral or generalised quaternion defect groups (see [[References|[Er90] ]]) and in the cases of dihedral and semihedral defect groups determined which are realised by blocks of finite groups. In the case of generalised quaternion groups, the case of blocks with two simple modules is still open. These classifications only hold with respect to the field ''k'' at present.<br />
<br />
Principal blocks with dihedral defect groups are classified up to source algebra equivalence in [[References#K|[KoLa20]]].<br />
<br />
== Abelian <math>2</math>-groups with <math>2</math>-rank at most three ==<br />
<br />
These have been classified in [[References#W|[WZZ18]]] and [[References#E|[EL18a]]] with respect to <math>\mathcal{O}</math>. The derived equivalences classes with respect to <math>\mathcal{O}</math> are known.<br />
<br />
Let <math>l,m,n \geq 1</math> be distinct with <math>l,m \neq 1</math><br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>rk_2(D) \leq 3</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>D</math> <br />
! scope="col"| # <math>(\mathcal{O}</math>-)Morita classes<br />
! scope="col"| # <math>\mathcal{O}</math>-Derived equiv classes<br />
! scope="col"| Always endopermutation source Morita equivalent?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|[[C(2^n)|<math>C_{2^n}</math>]] || 1 || 1 || Yes || [[References#L|[Li96]]] || [[Blocks with cyclic defect groups]]<br />
|-<br />
|[[C2xC2|<math>C_2 \times C_2</math>]] || 3 || 2 || Yes || [[References#L|[Li94]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)|<math>C_{2^m} \times C_{2^m}</math>]] || 2 || 2 || Yes || [[References#E|[EKKS14]]] ||<br />
|-<br />
|[[C(2^m)xC(2^n)|<math>C_{2^m} \times C_{2^n}</math>]] || 1 || 1 || Yes || [[References#P|[Pu88]]] || [[Nilpotent blocks]]<br />
|-<br />
|[[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]] || 8 || 4 || || [[References#E|[Ea16]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)xC(2^m)|<math>C_{2^m} \times C_{2^m} \times C_{2^m}</math>]] || 4 || 4 || Yes || [[References#W|[WZZ18]]] ||<br />
|-<br />
|[[C(2^m)xC2xC2|<math>C_{2^m} \times C_2 \times C_2</math>]] || 3 || 2 || || [[References#E|[EL18a]]] ||<br />
|-<br />
|[[C(2^m)xC(2^m)xC(2^n)|<math>C_{2^m} \times C_{2^m} \times C_{2^n}</math>]] || 2 || 2 || || [[References#W|[WZZ18]]] ||<br />
|-<br />
|[[C(2^l)xC(2^m)xC(2^n)|<math>C_{2^l} \times C_{2^m} \times C_{2^n}</math>]] || 1 || 1 || Yes || [[References#P|[Pu88]]] || [[Nilpotent blocks]]<br />
|}<br />
<br />
<!--== Abelian <math>2</math>-groups ==<br />
<br />
Donovan's conjecture holds for ''2''-blocks with abelian defect groups. Some generic classification results are known for certain inertial quotients. These will be detailed here.--><br />
<br />
== Minimal nonabelian <math>2</math>-groups ==<br />
<br />
Blocks with defect groups which are minimal nonabelian <math>2</math>-groups of the form <math>P=\langle x,y:x^{2^r}=y^{2^r}=[x,y]^2=[x,[x,y]]=[y,[x,y]]=1 \rangle</math> are classified in [[References#E|[EKS12]]]. There are two <math>\mathcal{O}</math>-Morita equivalence classes, with representatives <math>\mathcal{O}P</math> and <math>\mathcal{O}(P:C_3)</math>.<br />
<br />
For arbitrary minimal nonabelian <math>2</math>-groups, by [[References#S|[Sa16]]] blocks with such defect groups and the same fusion system are isotypic.<br />
<br />
== Homocyclic <math>2</math>-groups when inertial quotient contains a Singer cycle == <br />
<br />
A Singer cycle is an element of order <math>p^n-1</math> in <math>\operatorname{Aut}({C_p}^n) \cong GL_n(p)</math>, and a subgroup generated by a Singer cycle acts transitively on the non-trivial elements of <math>(C_p)^n</math>.<br />
<br />
<math>2</math>-blocks with homocyclic defect group <math> D \cong (C_{2^m})^n </math> whose inertial quotient <math> \mathbb{E} </math> contains a Singer cycle are classified in [[References#E|[McK19]]]. In this situation, <math>\mathbb{E}</math> has the form <math>E:F </math> where <math>E \cong C_{2^n-1}</math> and <math>F</math> is trivial or a subgroup of <math>C_n</math>. There are three <math> \mathcal{O} </math>-Morita equivalence classes when <math>m=1, n =3 </math>; two when <math>m=1 , n \neq 3 </math>; and only one when <math>m> 1 </math>. The three classes have representatives [[M(8,5,7) | <math> \mathcal{O}J_1 </math>]], which occurs only when <math>m=1, n=3</math>; <math> \mathcal{O}(SL_2(2^n):F) </math>, which occurs only when <math>m=1 </math>; and <math> \mathcal{O}(D : \mathbb{E})</math>. <br />
<br />
The Morita equivalence between the block and the class representative is known to be basic, possibly except when <math>m=1,n=3</math>, since the Morita equivalences between the [[M(8,5,8) |principal block of <math>{\rm \operatorname{Aut}(SL_2(8))}</math>]] and the blocks [[M(8,5,8)#Other_notatable_representatives |<math>B_0(\mathcal{O}({}^2G_2(3^{2m+1})))</math>]] are not known to be basic.<br />
<br />
<br />
== Abelian <math>2</math>-groups when inertial quotient is cyclic and acts freely on the defect group == <br />
<br />
<math>2</math>-blocks with abelian defect group <math> D </math> whose inertial quotient <math> E </math> is a cyclic group that acts freely on the defect group (i.e. such that the stabiliser in <math> E </math> of any nontrivial element of <math>D</math> is trivial) are classified in [[References#E|[ArMcK20]]].<br />
<br />
If the action of <math>E</math> is transitive, then <math>D</math> is homocyclic, <math>E</math> is a Singer cycle and, by the previous section, the block is either Morita equivalent to the principal block of <math> \mathcal{O}SL_2(2^n) </math>, or to <math> \mathcal{O}(D : E)</math>. <br />
<br />
If the action of <math>E</math> is not transitive, then the block is Morita equivalent to <math> \mathcal{O}(D : E)</math>. <br />
<br />
In each case, the Morita equivalence between the block and the class representative is known to be basic.</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=References&diff=1200
References
2022-08-05T13:45:17Z
<p>Charles Eaton: [Th93]</p>
<hr />
<div>[[#A|A,]] [[#B|B,]] [[#C|C,]] [[#D|D,]] [[#E|E,]] [[#F|F,]] [[#G|G,]] [[#H|H,]] [[#I|I,]] [[#J|J,]] [[#K|K,]] [[#L|L,]] [[#M|M,]] [[#N|N,]] [[#O|O,]] [[#P|P,]] [[#Q|Q,]] [[#R|R,]] [[#S|S,]] [[#T|T]] [[#U|U,]] [[#V|V,]] [[#W|W,]] [[#X|X,]] [[#Y|Y,]] [[#Z|Z,]] <br />
<br />
{| <br />
|- id="A"<br />
|[Al79] || '''J. L. Alperin''', ''Projective modules for <math>SL(2,2^n)</math>'', J. Pure and Applied Algebra '''15''' (1979), 219-234.<br />
|-<br />
|[Al80] || '''J. L. Alperin''', ''Local representation theory'', The Santa Cruz Conference on Finite Groups., Proc. Sympos. Pure Math. '''37''' (1980), 369-375.<br />
|-<br />
|[AE81] || '''J. L. Alperin and L. Evens''', ''Representations, resoluutions and Quillen's dimension theorem'', J. Pure Appl. Algebra '''22''' (1981), 1-9.<br />
|-<br />
|[AE04] || '''Jianbei An and C. W. Eaton''', ''Blocks with trivial intersection defect groups'', Math. Z. '''247''' (2004), 461-486.<br />
|-<br />
|[Ar19] || '''C. G. Ardito''', [https://arxiv.org/abs/1908.02652 ''Morita equivalence classes of blocks with elementary abelian defect groups of order 32''], J. Algebra '''573''' (2021), 297-335.<br />
|-<br />
|[ArMcK20] || '''C. G. Ardito and E. McKernon''', ''[https://arxiv.org/abs/2010.08329 ''2-blocks with an abelian defect group and a freely acting cyclic inertial quotient''], [https://arxiv.org/abs/2010.08329 arxiv.org/abs/2010.08329]<br />
|-<br />
|[AS20] || '''C. G. Ardito and B. Sambale''', [https://www.iazd.uni-hannover.de/fileadmin/iazd/sambale/pdfs/Broue32.pdf ''Broué's Conjecture for 2-blocks with elementary abelian defect groups of order 32''], [https://www.iazd.uni-hannover.de/fileadmin/iazd/sambale/pdfs/Broue32.pdf Preprint.] <br />
|-<br />
|[AKO11] || '''M. Aschbacher, R. Kessar and B. Oliver''', ''Fusion systems in algebra and topology'', London Mathematical Society Lecture Notes '''391''', Cambridge University Press (2011).<br />
|- id="B"<br />
|[BK07] || '''D. Benson and R. Kessar''', ''Blocks inequivalent to their Frobenius twists'', J. Algebra '''315''' (2007), 588-599.<br />
|-<br />
|[BKL18] || '''R. Boltje, R. Kessar, and M. Linckelmann''', [https://doi.org/10.1016/j.jalgebra.2019.02.045 ''On Picard groups of blocks of finite groups''], J. Algebra '''558''' (2020), 70-101.<br />
|-<br />
|[Bra41] || '''R. Brauer''', ''Investigations on group characters'', Ann. Math. '''42''' (1941), 936-958.<br />
|-<br />
|[BP80] || '''M. Broué and L. Puig''', ''A Frobenius theorem for blocks'', Invent. Math. '''56''' (1980), 117-128.<br />
|-<br />
|[BP80b] || '''M. Broué and L. Puig''', ''Characters and local structure in G-algebras'', J. Algebra '''63''' (1980), 306-317.<br />
|- id="C"<br />
|[Cr11] || '''D. A. Craven''', ''The Theory of Fusion Systems: An Algebraic Approach'', Cambridge University Press (2011).<br />
|-<br />
|[Cr12] || '''D. A. Craven''', [https://arxiv.org/abs/1207.0116 ''Perverse Equivalences and Broué's Conjecture II: The Cyclic Case''], [https://arxiv.org/abs/1207.0116 arXiv:1207.0116]<br />
|-<br />
|[CDR18] || '''D. A. Craven, O. Dudas and R. Rouquier''', [https://arxiv.org/abs/1701.07097 ''The Brauer trees of unipotent blocks''], to appear, J. EMS, [https://arxiv.org/abs/1701.07097 arXiv:1701.07097] <br />
|-<br />
|[CEKL11] || '''D. A. Craven, C. W. Eaton, R. Kessar and M. Linckelmann''', ''The structure of blocks with a Klein four defect group'', Math. Z. '''268''' (2011), 441-476.<br />
|-<br />
|[CG12] || '''D. A. Craven and A. Glesser''', ''Fusion systems on small p-groups'', Trans. AMS '''364''' (2012) 5945-5967.<br />
|-<br />
|[CR13] || '''D. A. Craven and R. Rouquier''', ''Perverse equivalences and Broué's conjecture'', Adv. Math. '''248''' (2013), 1-58.<br />
|-<br />
|[CuRe81a] || '''C. W. Curtis and I. Reiner''', ''Methods of representation theory, with applications to finite groups and orders, Volume I'', Wiley-Interscience (1981).<br />
|-<br />
|[CuRe81b] || '''C. W. Curtis and I. Reiner''', ''Methods of representation theory, with applications to finite groups and orders, Volume II'', Wiley-Interscience (1981).<br />
|- id="D"<br />
|[Da66] || '''E. C. Dade''', ''Blocks with cyclic defect groups'', Ann. Math. '''84''' (1966), 20-48. <br />
|-<br />
|[DE20] || '''S. Danz and K. Erdmann''', [https://arxiv.org/abs/2008.10999 ''On Ext-Quivers of Blocks of weight two for symmetric groups''], [https://arxiv.org/abs/2008.10999 arXiv:2008.10999]<br />
|-<br />
|[Du14] || '''O. Dudas''', [https://arxiv.org/abs/1011.5478 ''Coxeter orbits and Brauer trees II''], Int. Math. Res. Not. '''15''' (2014), 4100-4123.<br />
|-<br />
|[Dü04] || '''O. Düvel''', ''On Donovan's conjecture'', J. Algebra '''272''' (2004), 1-26.<br />
|- id="E"<br />
|[Ea16] || '''C. W. Eaton''', ''Morita equivalence classes of <math>2</math>-blocks of defect three'', Proc. AMS '''144''' (2016), 1961-1970.<br />
|-<br />
|[Ea18] || '''C. W. Eaton''', [https://arxiv.org/abs/1612.03485 ''Morita equivalence classes of blocks with elementary abelian defect groups of order 16''], [https://arxiv.org/abs/1612.03485 arXiv:1612.03485]<br />
|-<br />
|[EEL18] || '''C. W. Eaton, F. Eisele and M. Livesey''', [https://arxiv.org/abs/1809.08152 ''Donovan's conjecture, blocks with abelian defect groups and discrete valuation rings''], Math. Z. '''295''' (2020), 249-264.<br />
|-<br />
|[EKKS14] || '''C. W. Eaton, R. Kessar, B. Külshammer and B. Sambale''', ''<math>2</math>-blocks with abelian defect groups'', Adv. Math. '''254''' (2014), 706-735.<br />
|-<br />
|[EKS12] || '''C. W. Eaton, B. Külshammer and B. Sambale''', ''<math>2</math>-blocks with minimal nonabelian defect groups, II'', J. Group Theory '''15''' (2012), 311-321.<br />
|-<br />
|[EL18a] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/1709.04331 Classifying blocks with abelian defect groups of rank 3 for the prime 2]'', J. Algebra '''515''' (2018), 1-18.<br />
|-<br />
|[EL18b] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/1803.03539 Donovan's conjecture and blocks with abelian defect groups]'', Proc. AMS. '''147''' (2019), 963-970.<br />
|-<br />
|[EL18c] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/1810.10950 Some examples of Picard groups of blocks]'', J. Algebra '''558''' (2020), 350-370.<br />
|-<br />
|[EL20] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/2006.11173 Donovan's conjecture and extensions by the centralizer of a defect group]'', J. Algebra '''582''' (2021), 157-176.<br />
|-<br />
|[Ei16] || '''F. Eisele''', ''Blocks with a generalized quaternion defect group and three simple modules over a <math>2</math>-adic ring'', J. Algebra '''456''' (2016), 294-322.<br />
|-<br />
|[Ei18] || '''F. Eisele''', ''[https://arxiv.org/abs/1807.05110 The Picard group of an order and Külshammer reduction]'', Algebr. Represent. Theory '''24''' (2021), 505-518.<br />
|-<br />
|[Ei19] || '''F. Eisele''', ''[https://arxiv.org/abs/1908.00129 On the geometry of lattices and finiteness of Picard groups]'', [https://arxiv.org/abs/1908.00129 arXiv:1908.00129]<br />
|-<br />
|[EiLiv20] || '''F. Eisele and M. Livesey''', ''[https://arxiv.org/abs/2006.13837 Arbitrarily large Morita Frobenius numbers]'', [https://arxiv.org/abs/2006.13837 arXiv:2006.13837]<br />
|-<br />
|[Er82] || '''K. Erdmann''', ''Blocks whose defect groups are Klein four groups: a correction'', J. Algebra '''76''' (1982), 505-518.<br />
|-<br />
|[Er87] || '''K. Erdmann''', ''Algebras and dihedral defect groups'', Proc. LMS '''54''' (1987), 88-114.<br />
|-<br />
|[Er88a] || '''K. Erdmann''', ''Algebras and quaternion defect groups, I'', Math. Ann. '''281''' (1988), 545-560.<br />
|-<br />
|[Er88b] || '''K. Erdmann''', ''Algebras and quaternion defect groups, II'', Math. Ann. '''281''' (1988), 561-582. <br />
|-<br />
|[Er88c] || '''K. Erdmann''', ''Algebras and semidihedral defect groups I'', Proc. LMS '''57''' (1988), 109-150. <br />
|-<br />
|[Er90] || ''' K. Erdmann''', ''Blocks of tame representation type and related algebras'', Lecture Notes in Mathematics '''1428''', Springer-Verlag (1990).<br />
|-<br />
|[Er90b] || '''K. Erdmann''', ''Algebras and semidihedral defect groups II'', Proc. LMS '''60''' (1990), 123-165.<br />
|- id="F"<br />
|[Fa17] || '''N. Farrell''', ''On the Morita Frobenius numbers of blocks of finite reductive groups'', J. Algebra '''471''' (2017), 299-318.<br />
|-<br />
|[FK18] || '''N. Farrell and R. Kessar''', [https://arxiv.org/abs/1805.02015 ''Rationality of blocks of quasi-simple finite groups''], Represent. Theory '''23''' (2019), 325-349.<br />
|- id="G"<br />
|[GMdelR21] || '''D. Garcia, l. Margolis and A. del Rio''', [https://arxiv.org/abs/2016.07231 ''Non-isomorphic 2-groups with isomorphic modular group algebras''], [https://arxiv.org/abs/2016.07231 arXiv:2016.07231]<br />
|-<br />
|[GO97] || '''H. Gollan and T. Okuyama''', ''Derived equivalences for the smallest Janko group'', preprint (1997).<br />
|-<br />
|[GT19] || '''R. M. Guralnick and Pham Huu Tiep''', ''Sectional rank and Cohomology'', J. Algebra (2019) https://doi.org/10.1016/j.jalgebra.2019.04.023<br />
|- id="H"<br />
|[HM07] || '''G. T. Helleloid and U. Martin''', ''The automorphism group of a finite <math>p</math>-group is almost always a <math>p</math>-group'', J. Algebra (2007), 294-329.<br />
|-<br />
|[HP94] || '''H-W. Henn and S. Priddy''', ''<math>p</math>-nilpotence, classifying space indecompsability, and other properties of almost finite groups'', Comment. Math. Helvetici (1994), 335-350.<br />
|- <br />
|[HK00] || '''G. Hiss and R. Kessar''', ''Scopes reduction and Morita equivalence classes of blocks in finite classical groups'', J. Algebra '''230''' (2000), 378-423.<br />
|-<br />
|[HK05] || '''G. Hiss and R. Kessar''', ''Scopes reduction and Morita equivalence classes of blocks in finite classical groups II'', J. Algebra '''283''' (2005), 522-563.<br />
|-<br />
|[Ho97] || '''T. Holm''', ''Derived equivalent tame blocks'', J. Algebra '''194''' (1997), 178-200.<br />
|-<br />
|[HKL07] || '''T. Holm, R. Kessar and M. Linckelmann''', ''Blocks with a quaternion defect group over a 2-adic ring: the case <math>\tilde{A}_4</math>'', Glasgow Math. J. '''49''' (2007), 29–43.<br />
|- id="J"<br />
|[Ja69] || '''G. Janusz''', ''Indecomposable modules for finite groups'', Ann. Math. '''89''' (1969), 209-241.<br />
|-<br />
|[Jo96] || '''T. Jost''', ''Morita equivalences for blocks of finite general linear groups'', Manuscripta Math. '''91''' (1996), 121-144.<br />
|- id="K"<br />
|[Ke96] || '''R. Kessar''', ''Blocks and source algebras for the double covers of the symmetric and alternating groups'', J. Algebra '''186''' (1996), 872-933.<br />
|-<br />
|[Ke00] || '''R. Kessar''', ''Equivalences for blocks of the Weyl groups'', Proc. Amer. Math. Soc. '''128''' (2000), 337-346.<br />
|-<br />
|[Ke01] || '''R. Kessar''', ''Source algebra equivalences for blocks of finite general linear groups over a fixed field'', Manuscripta Math. '''104''' (2001), 145-162. <br />
|-<br />
|[Ke02] || '''R. Kessar''', ''Scopes reduction for blocks of finite alternating groups'', Quart. J. Math. '''53''' (2002), 443-454.<br />
|-<br />
|[Ke05] || ''' R. Kessar''', ''A remark on Donovan's conjecture'', Arch. Math (Basel) '''82''' (2005), 391-394.<br />
|-<br />
|[KL18] || '''R. Kessar and M. Linckelmann''', [https://arxiv.org/abs/1705.07227 ''Descent of equivalences and character bijections''], [https://arxiv.org/abs/1705.07227 arXiv:1705.07227]<br />
|-<br />
|[Ki84] || '''M. Kiyota''', ''On 3-blocks with an elementary abelian defect group of order 9'', J. Fac. Sci. Univ. Tokyo Sect. IA Math. '''31''' (1984), 33–58.<br />
|-<br />
|[Ko03] || '''S. Koshitani''', ''Conjectures of Donovan and Puig for principal <math>3</math>-blocks with abelian defect groups'', Comm. Alg. '''31''' (2003), 2229-2243; ''Corrigendum'', '''32''' (2004), 391-393.<br />
|-<br />
|[KKW02] || '''S. Koshitani, N. Kunugi and K. Waki''', ''Broué's conjecture for non-principal 3-blocks of finite groups'', J. Pure and Applied Algebra '''173''' (2002), 177-211. <br />
|-<br />
|[KKW04] || '''S. Koshitani, N. Kunugi and K. Waki''', ''Broué's abelian defect group conjecture for Held group and the sporadic Suzuki group'', J. Algebra '''279''' (2004), 638-666. <br />
|-<br />
|[KoLa20] || '''S. Koshitani and C. Lassueur''', ''Splendid Morita equivalences for principal 2-blocks with dihedral defect groups'', Math. Z. '''294''' (2020), 639-666.<br />
|-<br />
|[KoLa20b] || '''S. Koshitani and C. Lassueur''', ''Splendid Morita equivalences for principal blocks with generalised quaternion defect groups'', J. Algebra '''558''' (2020), 523-533.<br />
|-<br />
|[Kü80] || '''B. Külshammer''', ''On 2-blocks with wreathed defect groups'', J. Algebra '''64''' (1980), 529–555.<br />
|-<br />
|[Kü81] || '''B. Külshammer''', ''On p-blocks of p-solvable groups'', Comm. Alg. '''9''' (1981), 1763-1785.<br />
|-<br />
|[Kü91] || '''B. Külshammer''', ''Group-theoretical descriptions of ring-theoretical invariants of group algebras'', in Representation Theory of Finite Groups and Finite-Dimensional Algebras (Bielefeld, 1991), Progr. Math. '''95''', pp. 425-442, Birkhauser (1991).<br />
|-<br />
|[Kü95] || '''B. Külshammer''', ''Donovan's conjecture, crossed products and algebraic group actions'', Israel J. Math. '''92''' (1995), 295-306.<br />
|-<br />
|[KS13] || '''B. Külshammer and B. Sambale''', ''The 2-blocks of defect 4'', Representation Theory '''17''' (2013), 226-236.<br />
|-<br />
|[Ku00] || '''N. Kunugi''', ''Morita equivalent 3-blocks of the 3-dimensional projective special linear groups'', Proc. LMS '''80''' (2000), 575-589.<br />
|-<br />
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|- id="L"<br />
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|-<br />
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|}</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Fusion-trivial_p-groups&diff=1199
Fusion-trivial p-groups
2022-08-05T13:42:15Z
<p>Charles Eaton: [HP94] reference</p>
<hr />
<div>A p-group <math>P</math> is ''p-nilpotent forcing'' if any finite group <math>G</math> that contains <math>P</math> as a Sylow p-subgroup must be p-nilpotent (that is <math>G=O_{p'}(G)P</math>). These groups appear in [[References#W|[vdW91]]].<br />
<br />
There does not seem to be any name given to p-groups <math>P</math> for which the only saturated fusion system is <math>\mathcal{F}_P(P)</math>. We will refer to them as ''fusion-trivial p-groups'' (although appropriate names might also be ''nilpotent forcing'' or ''fusion nilpotent forcing''). Examples of such p-groups are abelian 2-groups <math>P</math> for which <math>{\rm Aut}(P)</math> is a 2-group, i.e., those abelian 2-groups whose cyclic direct factors have pairwise distinct orders.<br />
<br />
A p-group <math>P</math> is fusion-trivial if and only if it is [[Glossary|resistant]] and <math>{\rm Aut}(P)</math> is a p-group. Recall that <math>P</math> is resistant if whenever <math>\mathcal{F}</math> is a saturated fusion system on <math>P</math>, we have <math>\mathcal{F}=N_{\mathcal{F}}(P)</math>, or equivalently <math>\mathcal{F}=\mathcal{F}_P(G)</math> for some finite group <math>G</math> with <math>P</math> as a normal Sylow p-subgroup. Resistant p-groups were introduced in [[References#S|[St02]]] in terms of fusion systems for groups, and for arbitrary saturated fusion systems in [[References#S|[St06]]].<br />
<br />
Theorem 4.8 of [[References#S|[St06]]] yields a useful necessary and sufficient condition for a p-group <math>P</math> to be resistant: there exists a central series \[P=P_n \geq P_{n-1} \geq \cdots \geq P_1\] for which each <math>P_i</math> is weakly <math>\mathcal{F}</math>-closed in any saturated fusion system on <math>P</math>. This happens for example if each <math>P_i</math> is the unique subgroup of <math>P</math> of its isomorphism type.<br />
<br />
Following [[References#M|[Ma86]]] (and [[References#H|[HM07]]]), Henn and Priddy proved in [[References#H|[HP94]]] that in some sense asymptotically most <math>p</math>-groups only occur as Sylow <math>p</math>-subgroups of <math>p</math>-nilpotent groups. In [[References#T|[Th93]]] proved that the <math>p</math>-groups considered in [[References#H|[HP94]]] have a strongly characteristic central series, in which each term is the unique subgroup of its isomorphism type. Hence in the sense of [[References#M|[Ma86]]], almost every <math>p</math>-group is fusion trivial. This leaves the natural question of whether a version of this result with a cleaner definition of "almost all" holds:<br />
<br />
<div class="boxed"><br />
=== Question on fusion-trivial p-groups ===<br />
Does the proportion of p-groups of order <math>p^n</math> that are fusion-trivial tend to 1 as <math>n \rightarrow \infty</math>? <br />
</div><br />
<br />
In practice, a more realistic question would mimic the asymptotic results mentioned above.</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=References&diff=1198
References
2022-08-05T13:19:22Z
<p>Charles Eaton: [Ma86]</p>
<hr />
<div>[[#A|A,]] [[#B|B,]] [[#C|C,]] [[#D|D,]] [[#E|E,]] [[#F|F,]] [[#G|G,]] [[#H|H,]] [[#I|I,]] [[#J|J,]] [[#K|K,]] [[#L|L,]] [[#M|M,]] [[#N|N,]] [[#O|O,]] [[#P|P,]] [[#Q|Q,]] [[#R|R,]] [[#S|S,]] [[#T|T]] [[#U|U,]] [[#V|V,]] [[#W|W,]] [[#X|X,]] [[#Y|Y,]] [[#Z|Z,]] <br />
<br />
{| <br />
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|[Ea18] || '''C. W. Eaton''', [https://arxiv.org/abs/1612.03485 ''Morita equivalence classes of blocks with elementary abelian defect groups of order 16''], [https://arxiv.org/abs/1612.03485 arXiv:1612.03485]<br />
|-<br />
|[EEL18] || '''C. W. Eaton, F. Eisele and M. Livesey''', [https://arxiv.org/abs/1809.08152 ''Donovan's conjecture, blocks with abelian defect groups and discrete valuation rings''], Math. Z. '''295''' (2020), 249-264.<br />
|-<br />
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|-<br />
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|-<br />
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|-<br />
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|-<br />
|[KL18] || '''R. Kessar and M. Linckelmann''', [https://arxiv.org/abs/1705.07227 ''Descent of equivalences and character bijections''], [https://arxiv.org/abs/1705.07227 arXiv:1705.07227]<br />
|-<br />
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|-<br />
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|-<br />
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|-<br />
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|-<br />
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|-<br />
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|-<br />
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|-<br />
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|-<br />
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|- id="L"<br />
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|-<br />
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|-<br />
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|-<br />
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|-<br />
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|-<br />
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|-<br />
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|-<br />
|[Liv19] || '''M. Livesey''', [https://arxiv.org/abs/1907.12167 ''On Picard groups of blocks with normal defect groups''], J. Algebra '''566''' (2021), 94-118.<br />
|-<br />
|[LiMa20] || '''M. Livesey and C. Marchi''', [https://arxiv.org/abs/2002.10571 ''On Picent for blocks with normal defect group''], [https://arxiv.org/abs/2002.10571 arXiv:2002.10571]<br />
|-<br />
|[LiMa20b] || '''M. Livesey and C. Marchi''', [https://arxiv.org/abs/2008.05857 ''Picard groups for blocks with normal defect groups and linear source bimodules''], [https://arxiv.org/abs/2008.05857 arXiv:2008.05857]<br />
|- id="M"<br />
|[Mac] || '''N. Macgregor''', ''Morita equivalence classes of tame blocks of finite groups'', J. Algebra '''608''' (2022), 719-754.<br />
|-<br />
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|-<br />
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|-<br />
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|- id="N"<br />
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|- <br />
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|- id="O"<br />
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|- id="P"<br />
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|-<br />
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|-<br />
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|-<br />
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|- id="R"<br />
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|-<br />
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|-<br />
|[Ru11] || '''P. Ruengrot''', ''Perfect isometry groups for blocks of finite groups'', PhD Thesis, University of Manchester (2011).<br />
|- id="S"<br />
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|-<br />
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|-<br />
|[Sa12b] || '''B. Sambale''', ''Fusion systems on metacyclic 2-groups'', Osaka J. Math. '''49''' (2012), 325–329.<br />
|-<br />
|[Sa13] || '''B. Sambale''', ''Blocks with defect group <math>Q_{2^n} \times C_{2^m}</math> and <math>SD_{2^n} \times C_{2^m}</math>'', Algebr. Represent. Theory '''16''' (2013), 1717–1732.<br />
|-<br />
|[Sa13b] || '''B. Sambale''', ''Blocks with central product defect group <math>D_{2^n} ∗ C_{2^m}</math>'', Proc. Amer. Math. Soc. '''141''' (2013), 4057–4069.<br />
|-<br />
|[Sa13c] || '''B. Sambale''', ''Further evidence for conjectures in block theory'', Algebra Number Theory '''7''' (2013), 2241–2273. <br />
|-<br />
|[Sa14] || '''B. Sambale''', ''Blocks of Finite Groups and Their Invariants'', Lecture Notes in Mathematics, Springer (2014).<br />
|-<br />
|[Sa16] || '''B. Sambale''', ''2-blocks with minimal nonabelian defect groups III'', Pacific J. Math. '''280''' (2016), 475–487.<br />
|-<br />
|[Sa20] || '''B. Sambale''', [https://arxiv.org/abs/2005.13172 ''Blocks with small-dimensional basic algebra''], Bul. Aust. Math. Soc. '''103''' (2021), 461-474.<br />
|-<br />
|[SSS98] || '''M. Schaps, D. Shapira and O. Shlomo''', ''Quivers of blocks with normal defect groups'', Proc. Symp. in Pure Mathematics '''63''', Amer. Math. Soc. (1998), 497-510.<br />
|-<br />
|[Sc91] || '''J. Scopes''', ''Cartan matrices and Morita equivalence for blocks of the symmetric groups'', J. Algebra '''142''' (1991), 441-455.<br />
|-<br />
|[Sh20] || '''V. Shalotenko''', ''Bounds on the dimension of Ext for finite groups'', J. Algebra '''550''' (2020), 266-289.<br />
|-<br />
|[St02] || '''R. Stancu''', ''Almost all generalized extraspecial p-groups are resistant'', J. Algebra '''249''' (2002), 120-126.<br />
|-<br />
|[St06] || '''R. Stancu''', ''Control of fusion in fusion systems'', J. Algebra and its Applications '''5''' (2006), 817-837. <br />
|- id="V"<br />
|[vdW91] || '''R. van der Waall''', ''On p-nilpotent forcing groups'', Indag. Mathem., N.S., '''2''' (1991), 367-384.<br />
|- id="W"<br />
|[Wa00]|| '''A. Watanabe''', ''A remark on a splitting theorem for blocks with abelian defect groups'', RIMS Kokyuroku '''1140''', Edited by H.Sasaki, Research Institute for Mathematical Sciences, Kyoto University (2000), 76-79.<br />
|-<br />
|[WZZ18] || '''Chao Wu, Kun Zhang and Yuanyang Zhou''', ''[https://arxiv.org/abs/1803.07262 Blocks with defect group <math>Z_{2^n} \times Z_{2^n} \times Z_{2^m}</math>]'', J. Algebra '''510''' (2018), 469-498.<br />
|}</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=References&diff=1197
References
2022-08-05T13:00:09Z
<p>Charles Eaton: [HM07] and [HP94]</p>
<hr />
<div>[[#A|A,]] [[#B|B,]] [[#C|C,]] [[#D|D,]] [[#E|E,]] [[#F|F,]] [[#G|G,]] [[#H|H,]] [[#I|I,]] [[#J|J,]] [[#K|K,]] [[#L|L,]] [[#M|M,]] [[#N|N,]] [[#O|O,]] [[#P|P,]] [[#Q|Q,]] [[#R|R,]] [[#S|S,]] [[#T|T]] [[#U|U,]] [[#V|V,]] [[#W|W,]] [[#X|X,]] [[#Y|Y,]] [[#Z|Z,]] <br />
<br />
{| <br />
|- id="A"<br />
|[Al79] || '''J. L. Alperin''', ''Projective modules for <math>SL(2,2^n)</math>'', J. Pure and Applied Algebra '''15''' (1979), 219-234.<br />
|-<br />
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|-<br />
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|-<br />
|[AE04] || '''Jianbei An and C. W. Eaton''', ''Blocks with trivial intersection defect groups'', Math. Z. '''247''' (2004), 461-486.<br />
|-<br />
|[Ar19] || '''C. G. Ardito''', [https://arxiv.org/abs/1908.02652 ''Morita equivalence classes of blocks with elementary abelian defect groups of order 32''], J. Algebra '''573''' (2021), 297-335.<br />
|-<br />
|[ArMcK20] || '''C. G. Ardito and E. McKernon''', ''[https://arxiv.org/abs/2010.08329 ''2-blocks with an abelian defect group and a freely acting cyclic inertial quotient''], [https://arxiv.org/abs/2010.08329 arxiv.org/abs/2010.08329]<br />
|-<br />
|[AS20] || '''C. G. Ardito and B. Sambale''', [https://www.iazd.uni-hannover.de/fileadmin/iazd/sambale/pdfs/Broue32.pdf ''Broué's Conjecture for 2-blocks with elementary abelian defect groups of order 32''], [https://www.iazd.uni-hannover.de/fileadmin/iazd/sambale/pdfs/Broue32.pdf Preprint.] <br />
|-<br />
|[AKO11] || '''M. Aschbacher, R. Kessar and B. Oliver''', ''Fusion systems in algebra and topology'', London Mathematical Society Lecture Notes '''391''', Cambridge University Press (2011).<br />
|- id="B"<br />
|[BK07] || '''D. Benson and R. Kessar''', ''Blocks inequivalent to their Frobenius twists'', J. Algebra '''315''' (2007), 588-599.<br />
|-<br />
|[BKL18] || '''R. Boltje, R. Kessar, and M. Linckelmann''', [https://doi.org/10.1016/j.jalgebra.2019.02.045 ''On Picard groups of blocks of finite groups''], J. Algebra '''558''' (2020), 70-101.<br />
|-<br />
|[Bra41] || '''R. Brauer''', ''Investigations on group characters'', Ann. Math. '''42''' (1941), 936-958.<br />
|-<br />
|[BP80] || '''M. Broué and L. Puig''', ''A Frobenius theorem for blocks'', Invent. Math. '''56''' (1980), 117-128.<br />
|-<br />
|[BP80b] || '''M. Broué and L. Puig''', ''Characters and local structure in G-algebras'', J. Algebra '''63''' (1980), 306-317.<br />
|- id="C"<br />
|[Cr11] || '''D. A. Craven''', ''The Theory of Fusion Systems: An Algebraic Approach'', Cambridge University Press (2011).<br />
|-<br />
|[Cr12] || '''D. A. Craven''', [https://arxiv.org/abs/1207.0116 ''Perverse Equivalences and Broué's Conjecture II: The Cyclic Case''], [https://arxiv.org/abs/1207.0116 arXiv:1207.0116]<br />
|-<br />
|[CDR18] || '''D. A. Craven, O. Dudas and R. Rouquier''', [https://arxiv.org/abs/1701.07097 ''The Brauer trees of unipotent blocks''], to appear, J. EMS, [https://arxiv.org/abs/1701.07097 arXiv:1701.07097] <br />
|-<br />
|[CEKL11] || '''D. A. Craven, C. W. Eaton, R. Kessar and M. Linckelmann''', ''The structure of blocks with a Klein four defect group'', Math. Z. '''268''' (2011), 441-476.<br />
|-<br />
|[CG12] || '''D. A. Craven and A. Glesser''', ''Fusion systems on small p-groups'', Trans. AMS '''364''' (2012) 5945-5967.<br />
|-<br />
|[CR13] || '''D. A. Craven and R. Rouquier''', ''Perverse equivalences and Broué's conjecture'', Adv. Math. '''248''' (2013), 1-58.<br />
|-<br />
|[CuRe81a] || '''C. W. Curtis and I. Reiner''', ''Methods of representation theory, with applications to finite groups and orders, Volume I'', Wiley-Interscience (1981).<br />
|-<br />
|[CuRe81b] || '''C. W. Curtis and I. Reiner''', ''Methods of representation theory, with applications to finite groups and orders, Volume II'', Wiley-Interscience (1981).<br />
|- id="D"<br />
|[Da66] || '''E. C. Dade''', ''Blocks with cyclic defect groups'', Ann. Math. '''84''' (1966), 20-48. <br />
|-<br />
|[DE20] || '''S. Danz and K. Erdmann''', [https://arxiv.org/abs/2008.10999 ''On Ext-Quivers of Blocks of weight two for symmetric groups''], [https://arxiv.org/abs/2008.10999 arXiv:2008.10999]<br />
|-<br />
|[Du14] || '''O. Dudas''', [https://arxiv.org/abs/1011.5478 ''Coxeter orbits and Brauer trees II''], Int. Math. Res. Not. '''15''' (2014), 4100-4123.<br />
|-<br />
|[Dü04] || '''O. Düvel''', ''On Donovan's conjecture'', J. Algebra '''272''' (2004), 1-26.<br />
|- id="E"<br />
|[Ea16] || '''C. W. Eaton''', ''Morita equivalence classes of <math>2</math>-blocks of defect three'', Proc. AMS '''144''' (2016), 1961-1970.<br />
|-<br />
|[Ea18] || '''C. W. Eaton''', [https://arxiv.org/abs/1612.03485 ''Morita equivalence classes of blocks with elementary abelian defect groups of order 16''], [https://arxiv.org/abs/1612.03485 arXiv:1612.03485]<br />
|-<br />
|[EEL18] || '''C. W. Eaton, F. Eisele and M. Livesey''', [https://arxiv.org/abs/1809.08152 ''Donovan's conjecture, blocks with abelian defect groups and discrete valuation rings''], Math. Z. '''295''' (2020), 249-264.<br />
|-<br />
|[EKKS14] || '''C. W. Eaton, R. Kessar, B. Külshammer and B. Sambale''', ''<math>2</math>-blocks with abelian defect groups'', Adv. Math. '''254''' (2014), 706-735.<br />
|-<br />
|[EKS12] || '''C. W. Eaton, B. Külshammer and B. Sambale''', ''<math>2</math>-blocks with minimal nonabelian defect groups, II'', J. Group Theory '''15''' (2012), 311-321.<br />
|-<br />
|[EL18a] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/1709.04331 Classifying blocks with abelian defect groups of rank 3 for the prime 2]'', J. Algebra '''515''' (2018), 1-18.<br />
|-<br />
|[EL18b] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/1803.03539 Donovan's conjecture and blocks with abelian defect groups]'', Proc. AMS. '''147''' (2019), 963-970.<br />
|-<br />
|[EL18c] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/1810.10950 Some examples of Picard groups of blocks]'', J. Algebra '''558''' (2020), 350-370.<br />
|-<br />
|[EL20] || '''C. W. Eaton and M. Livesey''', ''[https://arxiv.org/abs/2006.11173 Donovan's conjecture and extensions by the centralizer of a defect group]'', J. Algebra '''582''' (2021), 157-176.<br />
|-<br />
|[Ei16] || '''F. Eisele''', ''Blocks with a generalized quaternion defect group and three simple modules over a <math>2</math>-adic ring'', J. Algebra '''456''' (2016), 294-322.<br />
|-<br />
|[Ei18] || '''F. Eisele''', ''[https://arxiv.org/abs/1807.05110 The Picard group of an order and Külshammer reduction]'', Algebr. Represent. Theory '''24''' (2021), 505-518.<br />
|-<br />
|[Ei19] || '''F. Eisele''', ''[https://arxiv.org/abs/1908.00129 On the geometry of lattices and finiteness of Picard groups]'', [https://arxiv.org/abs/1908.00129 arXiv:1908.00129]<br />
|-<br />
|[EiLiv20] || '''F. Eisele and M. Livesey''', ''[https://arxiv.org/abs/2006.13837 Arbitrarily large Morita Frobenius numbers]'', [https://arxiv.org/abs/2006.13837 arXiv:2006.13837]<br />
|-<br />
|[Er82] || '''K. Erdmann''', ''Blocks whose defect groups are Klein four groups: a correction'', J. Algebra '''76''' (1982), 505-518.<br />
|-<br />
|[Er87] || '''K. Erdmann''', ''Algebras and dihedral defect groups'', Proc. LMS '''54''' (1987), 88-114.<br />
|-<br />
|[Er88a] || '''K. Erdmann''', ''Algebras and quaternion defect groups, I'', Math. Ann. '''281''' (1988), 545-560.<br />
|-<br />
|[Er88b] || '''K. Erdmann''', ''Algebras and quaternion defect groups, II'', Math. Ann. '''281''' (1988), 561-582. <br />
|-<br />
|[Er88c] || '''K. Erdmann''', ''Algebras and semidihedral defect groups I'', Proc. LMS '''57''' (1988), 109-150. <br />
|-<br />
|[Er90] || ''' K. Erdmann''', ''Blocks of tame representation type and related algebras'', Lecture Notes in Mathematics '''1428''', Springer-Verlag (1990).<br />
|-<br />
|[Er90b] || '''K. Erdmann''', ''Algebras and semidihedral defect groups II'', Proc. LMS '''60''' (1990), 123-165.<br />
|- id="F"<br />
|[Fa17] || '''N. Farrell''', ''On the Morita Frobenius numbers of blocks of finite reductive groups'', J. Algebra '''471''' (2017), 299-318.<br />
|-<br />
|[FK18] || '''N. Farrell and R. Kessar''', [https://arxiv.org/abs/1805.02015 ''Rationality of blocks of quasi-simple finite groups''], Represent. Theory '''23''' (2019), 325-349.<br />
|- id="G"<br />
|[GMdelR21] || '''D. Garcia, l. Margolis and A. del Rio''', [https://arxiv.org/abs/2016.07231 ''Non-isomorphic 2-groups with isomorphic modular group algebras''], [https://arxiv.org/abs/2016.07231 arXiv:2016.07231]<br />
|-<br />
|[GO97] || '''H. Gollan and T. Okuyama''', ''Derived equivalences for the smallest Janko group'', preprint (1997).<br />
|-<br />
|[GT19] || '''R. M. Guralnick and Pham Huu Tiep''', ''Sectional rank and Cohomology'', J. Algebra (2019) https://doi.org/10.1016/j.jalgebra.2019.04.023<br />
|- id="H"<br />
|[HM07] || '''G. T. Helleloid and U. Martin''', ''The automorphism group of a finite <math>p</math>-group is almost always a <math>p</math>-group'', J. Algebra (2007), 294-329.<br />
|-<br />
|[HP94] || '''H-W. Henn and S. Priddy''', ''<math>p</math>-nilpotence, classifying space indecompsability, and other properties of almost finite groups'', Comment. Math. Helvetici (1994), 335-350.<br />
|- <br />
|[HK00] || '''G. Hiss and R. Kessar''', ''Scopes reduction and Morita equivalence classes of blocks in finite classical groups'', J. Algebra '''230''' (2000), 378-423.<br />
|-<br />
|[HK05] || '''G. Hiss and R. Kessar''', ''Scopes reduction and Morita equivalence classes of blocks in finite classical groups II'', J. Algebra '''283''' (2005), 522-563.<br />
|-<br />
|[Ho97] || '''T. Holm''', ''Derived equivalent tame blocks'', J. Algebra '''194''' (1997), 178-200.<br />
|-<br />
|[HKL07] || '''T. Holm, R. Kessar and M. Linckelmann''', ''Blocks with a quaternion defect group over a 2-adic ring: the case <math>\tilde{A}_4</math>'', Glasgow Math. J. '''49''' (2007), 29–43.<br />
|- id="J"<br />
|[Ja69] || '''G. Janusz''', ''Indecomposable modules for finite groups'', Ann. Math. '''89''' (1969), 209-241.<br />
|-<br />
|[Jo96] || '''T. Jost''', ''Morita equivalences for blocks of finite general linear groups'', Manuscripta Math. '''91''' (1996), 121-144.<br />
|- id="K"<br />
|[Ke96] || '''R. Kessar''', ''Blocks and source algebras for the double covers of the symmetric and alternating groups'', J. Algebra '''186''' (1996), 872-933.<br />
|-<br />
|[Ke00] || '''R. Kessar''', ''Equivalences for blocks of the Weyl groups'', Proc. Amer. Math. Soc. '''128''' (2000), 337-346.<br />
|-<br />
|[Ke01] || '''R. Kessar''', ''Source algebra equivalences for blocks of finite general linear groups over a fixed field'', Manuscripta Math. '''104''' (2001), 145-162. <br />
|-<br />
|[Ke02] || '''R. Kessar''', ''Scopes reduction for blocks of finite alternating groups'', Quart. J. Math. '''53''' (2002), 443-454.<br />
|-<br />
|[Ke05] || ''' R. Kessar''', ''A remark on Donovan's conjecture'', Arch. Math (Basel) '''82''' (2005), 391-394.<br />
|-<br />
|[KL18] || '''R. Kessar and M. Linckelmann''', [https://arxiv.org/abs/1705.07227 ''Descent of equivalences and character bijections''], [https://arxiv.org/abs/1705.07227 arXiv:1705.07227]<br />
|-<br />
|[Ki84] || '''M. Kiyota''', ''On 3-blocks with an elementary abelian defect group of order 9'', J. Fac. Sci. Univ. Tokyo Sect. IA Math. '''31''' (1984), 33–58.<br />
|-<br />
|[Ko03] || '''S. Koshitani''', ''Conjectures of Donovan and Puig for principal <math>3</math>-blocks with abelian defect groups'', Comm. Alg. '''31''' (2003), 2229-2243; ''Corrigendum'', '''32''' (2004), 391-393.<br />
|-<br />
|[KKW02] || '''S. Koshitani, N. Kunugi and K. Waki''', ''Broué's conjecture for non-principal 3-blocks of finite groups'', J. Pure and Applied Algebra '''173''' (2002), 177-211. <br />
|-<br />
|[KKW04] || '''S. Koshitani, N. Kunugi and K. Waki''', ''Broué's abelian defect group conjecture for Held group and the sporadic Suzuki group'', J. Algebra '''279''' (2004), 638-666. <br />
|-<br />
|[KoLa20] || '''S. Koshitani and C. Lassueur''', ''Splendid Morita equivalences for principal 2-blocks with dihedral defect groups'', Math. Z. '''294''' (2020), 639-666.<br />
|-<br />
|[KoLa20b] || '''S. Koshitani and C. Lassueur''', ''Splendid Morita equivalences for principal blocks with generalised quaternion defect groups'', J. Algebra '''558''' (2020), 523-533.<br />
|-<br />
|[Kü80] || '''B. Külshammer''', ''On 2-blocks with wreathed defect groups'', J. Algebra '''64''' (1980), 529–555.<br />
|-<br />
|[Kü81] || '''B. Külshammer''', ''On p-blocks of p-solvable groups'', Comm. Alg. '''9''' (1981), 1763-1785.<br />
|-<br />
|[Kü91] || '''B. Külshammer''', ''Group-theoretical descriptions of ring-theoretical invariants of group algebras'', in Representation Theory of Finite Groups and Finite-Dimensional Algebras (Bielefeld, 1991), Progr. Math. '''95''', pp. 425-442, Birkhauser (1991).<br />
|-<br />
|[Kü95] || '''B. Külshammer''', ''Donovan's conjecture, crossed products and algebraic group actions'', Israel J. Math. '''92''' (1995), 295-306.<br />
|-<br />
|[KS13] || '''B. Külshammer and B. Sambale''', ''The 2-blocks of defect 4'', Representation Theory '''17''' (2013), 226-236.<br />
|-<br />
|[Ku00] || '''N. Kunugi''', ''Morita equivalent 3-blocks of the 3-dimensional projective special linear groups'', Proc. LMS '''80''' (2000), 575-589.<br />
|-<br />
|[Kup69] || '''H. Kupisch''', ''Unzerlegbare Moduln endlicher Gruppen mit zyklischer p-Sylow Gruppe'', Math. Z. '''108''' (1969), 77-104.<br />
|- id="L"<br />
|[LM80]||'''P. Landrock and G. O. Michler''', ''Principal 2-blocks of the simple groups of Ree type'', Trans. AMS '''260''' (1980), 83-111.<br />
|-<br />
|[Li94] || '''M. Linckelmann''', ''The source algebras of blocks with a Klein four defect group'', J. Algebra '''167''' (1994), 821-854.<br />
|-<br />
|[Li94b] || '''M. Linckelmann''', ''A derived equivalence for blocks with dihedral defect groups'', J. Algebra '''164''' (1994), 244-255. <br />
|-<br />
|[Li96] || '''M. Linckelmann''', ''The isomorphism problem for cyclic blocks and their source algebras'', Invent. Math. '''125''' (1996), 265-283.<br />
|-<br />
|[Li18] || '''M. Linckelmann''', [https://arxiv.org/abs/1805.08884 ''The strong Frobenius numbers for cyclic defect blocks are equal to one''], [https://arxiv.org/abs/1805.08884 arXiv:1805.08884]<br />
|-<br />
|[Li18b] || '''M. Linckelmann''', ''Finite-dimensional algebras arising as blocks of finite group algebras'', Contemporary Mathematics '''705''' (2018), 155-188.<br />
|-<br />
|[Li18c] || '''M. Linckelmann''', ''The block theory of finite group algebras, Volume 1'', London Math. Soc. Student Texts '''92''', Cambridge University Press (2018).<br />
|-<br />
|[Li18d] || '''M. Linckelmann''', ''The block theory of finite group algebras, Volume 2'', London Math. Soc. Student Texts '''92''', Cambridge University Press (2018).<br />
|-<br />
|[LM20] || '''M. Linckelmann and W. Murphy''', [https://arxiv.org/abs/2005.02223 ''A 9-dimensional algebra which is not a block of a finite group''], [https://arxiv.org/abs/2005.02223 arXiv:2005.02223]<br />
|-<br />
|[Liv19] || '''M. Livesey''', [https://arxiv.org/abs/1907.12167 ''On Picard groups of blocks with normal defect groups''], J. Algebra '''566''' (2021), 94-118.<br />
|-<br />
|[LiMa20] || '''M. Livesey and C. Marchi''', [https://arxiv.org/abs/2002.10571 ''On Picent for blocks with normal defect group''], [https://arxiv.org/abs/2002.10571 arXiv:2002.10571]<br />
|-<br />
|[LiMa20b] || '''M. Livesey and C. Marchi''', [https://arxiv.org/abs/2008.05857 ''Picard groups for blocks with normal defect groups and linear source bimodules''], [https://arxiv.org/abs/2008.05857 arXiv:2008.05857]<br />
|- id="M"<br />
|[Mac] || '''N. Macgregor''', ''Morita equivalence classes of tame blocks of finite groups'', J. Algebra '''608''' (2022), 719-754.<br />
<br />
|-<br />
|[Mar] || '''C. Marchi''', ''Picard groups for blocks'', PhD thesis, University of Manchester (2022)<br />
|-<br />
|[McK19] || '''E. McKernon''', [https://arxiv.org/abs/1912.03222 ''2-Blocks whose defect group is homocyclic and whose inertial quotient contains a Singer cycle''], J. Algebra '''563''' (2020), 30–48.<br />
|-<br />
|[MS08] || '''J. Müller and M. Schaps''', ''The Broué conjecture for the faithful 3-blocks of <math>4.M_{22}</math>'', J. Algebra '''319''' (2008), 3588-3602.<br />
|- id="N"<br />
|[NS18] || '''G. Navarro and B. Sambale''', ''On the blockwise modular isomorphism problem'', Manuscripta Math. '''157''' (2018), 263-278.<br />
|- <br />
|[Ne02] || '''G. Nebe''', [http://www.math.rwth-aachen.de/~Gabriele.Nebe/papers/survey.pdf ''Group rings of finite groups over p-adic integers, some examples''], Proceedings of the conference Around Group rings (Edmonton) Resenhas '''5''' (2002), 329-350.<br />
|- id="O"<br />
|[Ok97] || '''T. Okuyama''', ''Some examples of derived equivalent blocks of finite groups'', preprint (1997).<br />
|- id="P"<br />
|[Pu88]|| '''L. Puig''', ''Nilpotent blocks and their source algebras'', Invent. Math. '''93''' (1988), 77-116.<br />
|-<br />
|[Pu94] || '''L. Puig''', ''On Joanna Scopes’ criterion of equivalence for blocks of symmetric groups'', Algebra Colloq. '''1''' (1994), 25-55.<br />
|-<br />
|[Pu99]|| '''L. Puig''', ''On the local structure of Morita and Rickard equivalences between Brauer blocks'', Progress in Math. '''178''', Birkhauser Verlag (1999).<br />
|-<br />
|[Pu09] || '''L. Puig''', ''Block source algebras in p-solvable groups'', Michigan Math. J. '''58''' (2009), 323-338.<br />
|- id="R"<br />
|[Ri96] || '''J. Rickard''', ''Splendid equivalences: derived categories and permutation modules'', Proc. London Math. Soc. '''72''' (1996), 331-358.<br />
|-<br />
|[Ro95] || '''R. Rouquier''', ''From stable equivalences to Rickard equivalences for blocks with cyclic defect'', Proceedings of Groups 1993, Galway-St. Andrews Conference, Vol. 2, London Math. Soc. Lecture Note Ser. '''212''', Cambridge University Press (1995), 512-523.<br />
|-<br />
|[Ru11] || '''P. Ruengrot''', ''Perfect isometry groups for blocks of finite groups'', PhD Thesis, University of Manchester (2011).<br />
|- id="S"<br />
|[Sa11] || '''B. Sambale''', ''<math>2</math>-blocks with minimal nonabelian defect groups'', J. Algebra '''337''' (2011), 261–284.<br />
|-<br />
|[Sa12] || '''B. Sambale''', ''Blocks with defect group <math>D_{2^n} \times C_{2^m}</math>'', J. Pure Appl. Algebra '''216''' (2012), 119–125.<br />
|-<br />
|[Sa12b] || '''B. Sambale''', ''Fusion systems on metacyclic 2-groups'', Osaka J. Math. '''49''' (2012), 325–329.<br />
|-<br />
|[Sa13] || '''B. Sambale''', ''Blocks with defect group <math>Q_{2^n} \times C_{2^m}</math> and <math>SD_{2^n} \times C_{2^m}</math>'', Algebr. Represent. Theory '''16''' (2013), 1717–1732.<br />
|-<br />
|[Sa13b] || '''B. Sambale''', ''Blocks with central product defect group <math>D_{2^n} ∗ C_{2^m}</math>'', Proc. Amer. Math. Soc. '''141''' (2013), 4057–4069.<br />
|-<br />
|[Sa13c] || '''B. Sambale''', ''Further evidence for conjectures in block theory'', Algebra Number Theory '''7''' (2013), 2241–2273. <br />
|-<br />
|[Sa14] || '''B. Sambale''', ''Blocks of Finite Groups and Their Invariants'', Lecture Notes in Mathematics, Springer (2014).<br />
|-<br />
|[Sa16] || '''B. Sambale''', ''2-blocks with minimal nonabelian defect groups III'', Pacific J. Math. '''280''' (2016), 475–487.<br />
|-<br />
|[Sa20] || '''B. Sambale''', [https://arxiv.org/abs/2005.13172 ''Blocks with small-dimensional basic algebra''], Bul. Aust. Math. Soc. '''103''' (2021), 461-474.<br />
|-<br />
|[SSS98] || '''M. Schaps, D. Shapira and O. Shlomo''', ''Quivers of blocks with normal defect groups'', Proc. Symp. in Pure Mathematics '''63''', Amer. Math. Soc. (1998), 497-510.<br />
|-<br />
|[Sc91] || '''J. Scopes''', ''Cartan matrices and Morita equivalence for blocks of the symmetric groups'', J. Algebra '''142''' (1991), 441-455.<br />
|-<br />
|[Sh20] || '''V. Shalotenko''', ''Bounds on the dimension of Ext for finite groups'', J. Algebra '''550''' (2020), 266-289.<br />
|-<br />
|[St02] || '''R. Stancu''', ''Almost all generalized extraspecial p-groups are resistant'', J. Algebra '''249''' (2002), 120-126.<br />
|-<br />
|[St06] || '''R. Stancu''', ''Control of fusion in fusion systems'', J. Algebra and its Applications '''5''' (2006), 817-837. <br />
|- id="V"<br />
|[vdW91] || '''R. van der Waall''', ''On p-nilpotent forcing groups'', Indag. Mathem., N.S., '''2''' (1991), 367-384.<br />
|- id="W"<br />
|[Wa00]|| '''A. Watanabe''', ''A remark on a splitting theorem for blocks with abelian defect groups'', RIMS Kokyuroku '''1140''', Edited by H.Sasaki, Research Institute for Mathematical Sciences, Kyoto University (2000), 76-79.<br />
|-<br />
|[WZZ18] || '''Chao Wu, Kun Zhang and Yuanyang Zhou''', ''[https://arxiv.org/abs/1803.07262 Blocks with defect group <math>Z_{2^n} \times Z_{2^n} \times Z_{2^m}</math>]'', J. Algebra '''510''' (2018), 469-498.<br />
|}</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Miscallaneous_results&diff=1196
Miscallaneous results
2022-08-04T13:51:27Z
<p>Charles Eaton: /* Morita invariance of the isomorphism type of a defect group */ Section rewritten following counterexample to modular isomorphism problem</p>
<hr />
<div>This page will contain results which do not fit in elsewhere on this site.<br />
<br />
== Blocks with basic algebras of low dimension ==<br />
[[Blocks with basic algebras of low dimension|Main article: Blocks with basic algebras of low dimension]]<br />
<br />
In [[References#L|[Li18b]]] Markus Linckelmann calculated the <math>k</math>-algebras of dimension at most twelve which occur as basic algebras of blocks of finite groups, with the exception of one case of dimension 9 where no block with that basic algebra is identified. This final case was ruled out by Linckelmann and Murphy in [[References#L|[LM20]]]. These results do not use the classification of finite simple groups. In [[References#S|[Sa20]]] Benjamin Sambale applied the classification of finite simple groups to extend the classification to dimensions 13 and 14. See [[Blocks with basic algebras of low dimension]] for a description of these results.<br />
<br />
== Morita (non-)invariance of the isomorphism type of a defect group ==<br />
[[Morita invariance of the isomorphism type of a defect group|Main article: Morita invariance of the isomorphism type of a defect group]]<br />
<br />
In [[References#G|[GMdelR21]]] examples are given of non-isomorphic <math>2</math>-groups whose group algebras over a field of characteristic <math>2</math> are isomorphic, thus giving a counterexample to the modular isomorphism problem for fields of prime characteristic. This gives examples of blocks with non-isomorphic defect groups that are Morita equivalent. Note that these do no yield examples of Morita equivalent blocks defined over a local ring, so the question is still open as to whether the defect group is an invariant under Morita equivalence of such blocks. <br />
<br />
Note that the examples in [[References#G|[GMdelR21]]] also yield blocks that are Morita equivalent but not via a [[Glossary#Basic Morita/stable equivalence|basic Morita equivalence]].</div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=SD16&diff=1195
SD16
2022-08-04T13:34:27Z
<p>Charles Eaton: /* Blocks with defect group SD_{16} */ Added note about Monster group</p>
<hr />
<div>__NOTITLE__<br />
<br />
== Blocks with defect group <math>SD_{16}</math> ==<br />
<br />
These are examples of [[Tame blocks|tame blocks]] and were first classified over <math>k</math> by Erdmann (see [[References|[Er88c], [Er90b]]]). Further work was carried out in [[References#M|[Mac]]], where <math>SD(3 {\cal H})</math> was eliminated, and the block <math>B_0(kPSL_3(3))</math> reattributed to <math>SD(3 {\cal D})</math>, so that the class <math>SD(3 {\cal B})_1</math> is now a class of algebras with no known block-realised representative. It is not known whether there are blocks realising the class <math>SD(3 {\cal C})_2</math>. <br />
<br />
Until these remaining cases are resolved the labelling is provisional.<br />
<br />
The classification with respect to <math>\mathcal{O}</math> is still unknown.<br />
<br />
'''<pre style="color: red">CLASSIFICATION INCOMPLETE</pre>'''<br />
<br />
{| class="wikitable"<br />
|-<br />
! scope="col"| Class<br />
! scope="col"| Representative<br />
! scope="col"| # lifts / <math>\mathcal{O}</math><br />
! scope="col"| <math>k(B)</math><br />
! scope="col"| <math>l(B)</math><br />
! scope="col"| Inertial quotients<br />
! scope="col"| <math>{\rm Pic}_\mathcal{O}(B)</math><br />
! scope="col"| <math>{\rm Pic}_k(B)</math><br />
! scope="col"| <math>{\rm mf_\mathcal{O}(B)}</math><br />
! scope="col"| <math>{\rm mf_k(B)}</math><br />
! scope="col"| Notes<br />
<br />
|-<br />
|[[M(16,8,1)]] || <math>kSD_{16}</math> || 1 ||7 ||1 ||<math>1</math> || || || ||1 ||<br />
|-<br />
|[[M(16,8,2)]] || <math>B_5(kPSU_3(5))</math> || ? ||8 ||2 ||<math>1</math> || || || ||1 || <math>SD(2 {\cal A})_1</math><br />
|-<br />
|[[M(16,8,3)]] || <math>B_0(kM_{10})=B_0(k(A_6.2_3))</math> || ? ||7 ||2 ||<math>1</math> || || || ||1 || <math>SD(2 {\cal A})_2</math><br />
|-<br />
|[[M(16,8,4)]] || <math>B_3(k(3.M_{10}))=B_3(k(3.A_6.2_3))</math> || ? ||7 ||2 ||<math>1</math> || || || ||1 || <math>SD(2 {\cal B})_1</math><br />
|-<br />
|[[M(16,8,5)]] || <math>B_1(kPSL_3(11))</math> || ? ||8 ||2 ||<math>1</math> || || || ||1 || <math>SD(2 {\cal B})_2</math><br />
|-<br />
|[[M(16,8,6)]] || <math>B_0(kPSU_3(5))</math> || ? ||8 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal A})_1</math><br />
|-<br />
|[[M(16,8,7)]] || <math>B_0(kPSL_3(3))</math> || ? ||8 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal D})</math>. See note above.<br />
|-<br />
| || || ? ||8 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal B})_1</math><br />
|-<br />
| || || ? ||8 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal B})_2</math> or <math>SD(3 {\cal C})_2</math><ref>See discussion in [[References#M|[Mac]]]: A block of the Monster group could be in one or other of these classes.</ref><br />
<br />
|}<br />
<br />
[[M(16,8,2)]] and [[M(16,8,5)]] are derived equivalent over <math>k</math> by [[References|[Ho97] ]].<br />
<br />
[[M(16,8,3)]] and [[M(16,8,4)]] are derived equivalent over <math>k</math> by [[References|[Ho97] ]].<br />
<br />
All Morita equivalence classes with three simple modules are derived equivalent over <math>k</math> by [[References|[Ho97] ]].<br />
<br />
<br />
== Notes ==<br />
<br />
<references /></div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=SD16&diff=1194
SD16
2022-08-04T13:23:27Z
<p>Charles Eaton: /* Blocks with defect group SD_{16} */ Removed M_{11}</p>
<hr />
<div>__NOTITLE__<br />
<br />
== Blocks with defect group <math>SD_{16}</math> ==<br />
<br />
These are examples of [[Tame blocks|tame blocks]] and were first classified over <math>k</math> by Erdmann (see [[References|[Er88c], [Er90b]]]). Further work was carried out in [[References#M|[Mac]]], where <math>SD(3 {\cal H})</math> was eliminated, and the block <math>B_0(kPSL_3(3))</math> reattributed to <math>SD(3 {\cal D})</math>, so that the class <math>SD(3 {\cal B})_1</math> is now a class of algebras with no known block-realised representative. It is not known whether there are blocks realising the class <math>SD(3 {\cal C})_2</math>. <br />
<br />
Until these remaining cases are resolved the labelling is provisional.<br />
<br />
The classification with respect to <math>\mathcal{O}</math> is still unknown.<br />
<br />
'''<pre style="color: red">CLASSIFICATION INCOMPLETE</pre>'''<br />
<br />
{| class="wikitable"<br />
|-<br />
! scope="col"| Class<br />
! scope="col"| Representative<br />
! scope="col"| # lifts / <math>\mathcal{O}</math><br />
! scope="col"| <math>k(B)</math><br />
! scope="col"| <math>l(B)</math><br />
! scope="col"| Inertial quotients<br />
! scope="col"| <math>{\rm Pic}_\mathcal{O}(B)</math><br />
! scope="col"| <math>{\rm Pic}_k(B)</math><br />
! scope="col"| <math>{\rm mf_\mathcal{O}(B)}</math><br />
! scope="col"| <math>{\rm mf_k(B)}</math><br />
! scope="col"| Notes<br />
<br />
|-<br />
|[[M(16,8,1)]] || <math>kSD_{16}</math> || 1 ||7 ||1 ||<math>1</math> || || || ||1 ||<br />
|-<br />
|[[M(16,8,2)]] || <math>B_5(kPSU_3(5))</math> || ? ||8 ||2 ||<math>1</math> || || || ||1 || <math>SD(2 {\cal A})_1</math><br />
|-<br />
|[[M(16,8,3)]] || <math>B_0(kM_{10})=B_0(k(A_6.2_3))</math> || ? ||7 ||2 ||<math>1</math> || || || ||1 || <math>SD(2 {\cal A})_2</math><br />
|-<br />
|[[M(16,8,4)]] || <math>B_3(k(3.M_{10}))=B_3(k(3.A_6.2_3))</math> || ? ||7 ||2 ||<math>1</math> || || || ||1 || <math>SD(2 {\cal B})_1</math><br />
|-<br />
|[[M(16,8,5)]] || <math>B_1(kPSL_3(11))</math> || ? ||8 ||2 ||<math>1</math> || || || ||1 || <math>SD(2 {\cal B})_2</math><br />
|-<br />
|[[M(16,8,6)]] || <math>B_0(kPSU_3(5))</math> || ? ||8 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal A})_1</math><br />
|-<br />
|[[M(16,8,7)]] || <math>B_0(kPSL_3(3))</math> || ? ||8 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal D})</math>. See note above.<br />
|-<br />
| || || ? ||8 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal B})_1</math><br />
|-<br />
| || || ? ||8 ||3 ||<math>1</math> || || || ||1 || <math>SD(3 {\cal C})_2</math><br />
<br />
|}<br />
<br />
[[M(16,8,2)]] and [[M(16,8,5)]] are derived equivalent over <math>k</math> by [[References|[Ho97] ]].<br />
<br />
[[M(16,8,3)]] and [[M(16,8,4)]] are derived equivalent over <math>k</math> by [[References|[Ho97] ]].<br />
<br />
All Morita equivalence classes with three simple modules are derived equivalent over <math>k</math> by [[References|[Ho97] ]].<br />
<br />
<!--[[M(8,3,4)]], [[M(8,3,5)]] and [[M(8,3,6)]] are derived equivalent over <math>k</math> by [[References|[Li94b] ]].--></div>
Charles Eaton
http://wiki.manchester.ac.uk/blocks/index.php?title=Classification_by_p-group&diff=1193
Classification by p-group
2022-08-04T13:21:23Z
<p>Charles Eaton: /* Blocks for p=2 */ Updated number of classes for SD_{16}</p>
<hr />
<div>'''Classification of Morita equivalences for blocks with a given defect group'''<br />
<br />
On this page we list classifications of Morita equivalence classes for each isomorphism class of p-groups in turn. [[Generic classifications by p-group class|Generic classifications for classes of p-groups can be found here]].<br />
<br />
See [[Labelling for Morita equivalence classes|this page]] for a description of the labelling conventions.<br />
<br />
== Blocks for <math> p=2 </math> ==<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>1 \leq |D| \leq 8</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|- <br />
| 2 || [[C2|1]] || [[C2|<math>C_2</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
| 4 || [[C4|1]] || [[C4|<math>C_4</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
| 4 || [[C2xC2|2]] || [[C2xC2|<math>C_2 \times C_2</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Er82], [Li94] ]] ||<br />
|- <br />
|8 || [[C8|1]] || [[C8|<math>C_8</math>]] ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|8 || [[C4xC2|2]] || [[C4xC2|<math>C_4 \times C_2</math>]] ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|8 || [[D8|3]] || [[D8|<math>D_8</math>]] ||6(?) || <math>k</math> || <math>k</math> || [[References|[Er87] ]] || Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] <br />
|-<br />
|8 || [[Q8|4]] || [[Q8|<math>Q_8</math>]] ||3(3) || <math>\mathcal{O}</math> || <math>k</math> || [[References|[Er88a], [Er88b], [HKL07], [Ei16]]] || <br />
|-<br />
|8 || [[C2xC2xC2|5]] || [[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]] || 8(8) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References| [Ea16]]] || Uses CFSG<br />
|}<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>|D|=16</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Donovan (w.r.t)?<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|16 || [[C16|1]] || [[C16|<math>C_{16}</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|16 || [[C4xC4|2]] || [[C4xC4|<math>C_4 \times C_4</math>]] || <math>\mathcal{O}</math> || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[EKKS14] ]] ||<br />
|-<br />
|16 || [[MNA(2,1)|3]] || [[MNA(2,1)]] || No || 3(?) || No || || [[References|[Sa11] ]] || Block invariants known<br />
|-<br />
|16 || [[C4:C4|4]] || [[C4:C4|<math>C_4:C_4</math>]] || <math>\mathcal{O}</math>|| 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[CG12], [Sa12b]]] ||<br />
|-<br />
|16 || [[C8xC2|5]] || [[C8xC2|<math>C_8 \times C_2</math>]]|| <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|16 || [[M16|6]] || [[M16|<math>M_{16}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[CG12], [Sa12b] ]] || <br />
|-<br />
|16 || [[D16|7]] || [[D16|<math>D_{16}</math>]] || <math>k</math>|| 5(?) || <math>k</math> || <math>k</math> || [[References|[Er87]]] || Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] <br />
|-<br />
|16 || [[SD16|8]] || [[SD16|<math>SD_{16}</math>]] || <math>k</math> || 7(?) || || || [[References|[Er88c], [Er90b]]] || Two other possible classes<br />
|-<br />
|16 || [[Q16|9]] || [[Q16|<math>Q_{16}</math>]] || No || 6(?) || || <math>k</math> || [[References|[Er88a], [Er88b], [Ho97]]] || Two possibly infinite families when <math>l(B)=2</math>. Classified over <math>\mathcal{O}</math> when <math>l(B)=3</math> in [[References#E|[Ei16]]]. Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20b]]]<br />
|-<br />
|16 || [[C4xC2xC2|10]] || [[C4xC2xC2|<math>C_4 \times C_2 \times C_2</math>]]|| <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[EL18a]]] ||<br />
|-<br />
|16 || [[D8xC2|11]] || [[D8xC2|<math>D_8 \times C_2</math>]] || No || || || || [[References|[Sa12] ]] || Block invariants known<br />
|-<br />
|16 || [[Q8xC2|12]] || [[Q8xC2|<math>Q_8 \times C_2</math>]] || <math>\mathcal{O}</math> || 3(3) || No || || [[References#E|[EL20]]] || Block invariants known by [[References#S|[Sa13]]]<br />
|-<br />
|16 || [[D8*C4|13]] || [[D8*C4|<math>D_8*C_4</math>]] || No || 3(?) || No || || [[References|[Sa13b] ]] || Block invariants known<br />
|-<br />
|16 || [[(C2)^4|14]] || [[(C2)^4|<math>(C_2)^4</math>]] || <math>\mathcal{O}</math> || 16(16) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Ea18] ]] ||<br />
|}<br />
<br />
The table for defect groups of order 32 takes as its starting point Table 13.1 of Sambale's book [[References|[Sa14]]].<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>|D|=32</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Donovan (w.r.t)?<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|32 || [[C32|1]] || [[C32|<math>C_{32}</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|32 || [[MNA(2,2)|2]] || [[MNA(2,2)|<math>MNA(2,2)</math>]] || <math>\mathcal{O}</math> || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EKS12]]] ||<br />
|-<br />
|32 || [[C8xC4|3]] || [[C8xC4|<math>C_8 \times C_4</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|-<br />
|32 || [[C8:C4|4]] || [[C8:C4|<math>C_8:C_4</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[MNA(3,1)|5]] || [[MNA(3,1)|<math>MNA(3,1)</math>]] || No || || || || [[References#S|[Sa11] ]] || Invariants known<br />
|-<br />
|32 || [[MNA(2,1):C2|6]] || [[MNA(3,1):C2|<math>MNA(2,1):C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,7)|7]] || [[SmallGroup(32,7)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] || <math>M_{16}:C_2</math><br />
|-<br />
|32 || [[2.MNA(2,1)|8]] || [[2.MNA(2,1)|<math>2.MNA(2,1)</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[D8:C4|9]] || [[D8:C4|<math>D_8:C_4</math>]] || No || || || || [[References#S|[Sa14,10.23]]] || Invariants known<br />
|-<br />
|32 || [[Q8:C4|10]] || [[Q8:C4|<math>Q_8:C_4</math>]] || No || || || || [[References#S|[Sa14,10.25]]] || Invariants known<br />
|-<br />
|32 || [[C4wrC2|11]] || [[C4wrC2|<math>C_4 \wr C_2</math>]] || No || || || || [[References#K|[Ku80]]] || Invariants known<br />
|-<br />
|32 || [[C4:C8|12]] || [[C4:C8|<math>C_4:C_8</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[C8:C4a|13]] || [[C8:C4a|<math>C_8:C_4=\langle a,b|a^8=b^4=1, ba=a^3b \rangle</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[C8:C4b|14]] || [[C8:C4b|<math>C_8:C_4=\langle a,b|a^8=b^4=1, ba=a^7b \rangle</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,15)|15]] || [[SmallGroup(32,15)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[C16xC2|16]] || [[C16xC2|<math>C_{16} \times C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|32 || [[M32|17]] || [[M32|<math>M_{32}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] ||<br />
|-<br />
|32 || [[D32|18]] || [[D32|<math>D_{32}</math>]] || <math>k</math> || 5(?) || <math>k</math> || <math>k</math> || [[References#E|[Er87]]] || Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] <br />
|-<br />
|32 || [[SD32|19]] || [[SD32|<math>SD_{32}</math>]] || <math>k</math> || || || || ||<br />
|-<br />
|32 || [[Q32|20]] || [[Q32|<math>Q_{32}</math>]] || No || || || || [[References#E|[Er88a], [Er88b], [Ho97]]] || Two possibly infinite families when <math>l(B)=2</math>. Classified over <math>\mathcal{O}</math> when <math>l(B)=3</math> in [[References#E|[Ei16]]]. Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20b]]]<br />
|-<br />
|32 || [[C4xC4xC2|21]] || [[C4xC4xC2|<math>C_4 \times C_4 \times C_2</math>]] || <math>\mathcal{O}</math> || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EKKS14]]] <br />
|-<br />
|32 || [[MNA(2,1)xC2|22]] || [[MNA(2,1)xC2|<math>MNA(2,1) \times C_2</math>]] || No || || || || [[References#S|[Sa14,10.25]]] || Invariants known<br />
|-<br />
|32 || [[(C4:C4)xC2|23]] || [[(C4:C4)xC2|<math>(C_4:C_4) \times C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,24)|24]] || [[SmallGroup(32,24)]]<!--<math>(C_4 \times C_4):C_2=\langle a,b,c \mid a^4 = b^4 = c^2 = e, ab = ba, ac = ca, cb = a^2bc \rangle</math>]]--> || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[D8xC4|25]] || [[D8xC4|<math>D_8 \times C_4</math>]] || No || || || || [[References#S|[Sa14,9.7]]] ||<br />
Invariants known<br />
|-<br />
|32 || [[Q8xC4|26]] || [[Q8xC4|<math>Q_8 \times C_4</math>]] || <math>\mathcal{O}</math> || 3(3) || No || || [[References#E|[EL20]]] || Invariants known by [[References#S|[Sa14,9.28]]]<br />
|-<br />
|32 || [[SmallGroup(32,27)|27]] || [[SmallGroup(32,27)]]<!--|<math>(C_4 \times C_4):C_2=\langle x,y,z,a,b \mid x^2 = y^2 = z^2 = a^2 = b^2 = e, xy = yx, xz, = zx, yz = zy, aza^{-1} = xz, bzb^{-1} = yz, ax = xa, ay = ya, bx = xb, by = yb \rangle</math>]]--> || No || || || || ||<br />
|-<br />
|32 || [[SmallGroup(32,28)|28]] || [[SmallGroup(32,28)]] || No || || || || [[References#S|[Sa14,13.11]]] || Invariants known<br />
|-<br />
|32 || [[SmallGroup(32,29)|29]] || [[SmallGroup(32,29)]] || No || || || || [[References#S|[Sa14,13.11]]] || Invariants known<br />
|-<br />
|32 || [[SmallGroup(32,30)|30]] || [[SmallGroup(32,30)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,31)|31]] || [[SmallGroup(32,31)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,32)|32]] || [[SmallGroup(32,32)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[SmallGroup(32,33)|33]] || [[SmallGroup(32,33)]] || No || || || || [[References|[Sa14,13.12]]] || Invariants partly known<br />
|-<br />
|32 || [[SmallGroup(32,34)|34]] || [[SmallGroup(32,34)]] || No || || || || [[References|[Sa14,13.12]]] || Invariants partly known<br />
|-<br />
|32 || [[C4:Q8|35]] || [[C4:Q8|<math>C_4:Q_8</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[C8xC2xC2|36]] || [[C8xC2xC2|<math>C_8 \times C_2 \times C_2</math>]] || <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EL18a]]] ||<br />
|-<br />
|32 || [[M16xC2|37]] || [[M16xC2|<math>M_{16} \times C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#S|[Sa14]]] ||<br />
|-<br />
|32 || [[D8*C8|38]] || [[D8*C8|<math>D_8*C_8</math>]] || No || || || || [[References#S|[Sa14,9.18]]] || Invariants known<br />
|-<br />
|32 || [[D16xC2|39]] || [[D16xC2|<math>D_{16} \times C_2</math>]] || No || || || || [[References#S|[Sa14,9.7]]] || Invariants known<br />
|-<br />
|32 || [[SD16xC2|40]] || [[SD16xC2|<math>SD_{16} \times C_2</math>]] || No || || || || [[References#S|[Sa14,9.37]]] || Invariants known<br />
|-<br />
|32 || [[Q16xC2|41]] || [[Q16xC2|<math>Q_{16} \times C_2</math>]] || No || || || || [[References#S|[Sa14,9.28]]] || Invariants known<br />
|-<br />
|32 || [[D16*C4|42]] || [[D16*C4|<math>D_{16}*C_4</math>]] || No || || || || [[References|[Sa14,9.18]]] || Invariants known<br />
|-<br />
|32 || [[SmallGroup(32,43)|43]] || [[SmallGroup(32,43)]] || No || || || || ||<br />
|-<br />
|32 || [[SmallGroup(32,44)|44]] || [[SmallGroup(32,44)]] || No || || || || ||<br />
|-<br />
|32 || [[C4xC2xC2xC2|45]] || [[C4xC2xC2xC2|<math>C_4 \times C_2 \times C_2 \times C_2</math>]] || <math>\mathcal{O}</math> || || || || [[References#S|[Sa14, 13.9]]] || Invariants known<br />
|-<br />
|32 || [[D8xC2xC2|46]] || [[D8xC2xC2|<math>D_8 \times C_2 \times C_2</math>]] || No || || || || ||<br />
|-<br />
|32 || [[Q8xC2xC2|47]] || [[Q8xC2xC2|<math>Q_8 \times C_2 \times C_2</math>]] || No || || || || ||<br />
|-<br />
|32 || [[D8*C4xC2|48]] || [[D8*C4xC2|<math>(D_8*C_4) \times C_2</math>]] || No || || || || ||<br />
|-<br />
|32 || [[D8*D8|49]] || [[D8*D8|<math>D_8*D_8</math>]] || No || || || || [[References#S|[Sa13c]]] || Invariants partly known<br />
|-<br />
|32 || [[D8*Q8|50]] || [[D8*Q8|<math>D_8*Q_8</math>]] || No || || || || [[References#S|[Sa13c]]] || Invariants partly known<br />
|-<br />
|32 || [[(C2)^5|51]] || [[(C2)^5|<math>(C_2)^5</math>]] || <math>\mathcal{O}</math> || 34 (34) || <math>\mathcal{O}</math> || || [[References#A|[Ar19]]] || Derived eq. classes determined for 30 of the 34 Morita eq. classes. <br />
|}<br />
<br />
<!--<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>|D|=64</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Donovan (w.r.t)?<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
|64 || [[C64|1]] || [[C64|<math>C_{64}</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
|64 || [[C8xC8|2]] || [[C8xC8|<math>C_8 \times C_8</math>]] || <math>\mathcal{O}</math> ||2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#E|[EKKS14]]]||<br />
|-<br />
|64 || [[SmallGroup(64,3)|3]] || [[SmallGroup(64,3)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_8:C_8</math><br />
|-<br />
|64 || [[(C2xC2xC2):C8|4]] || [[(C2xC2xC2):C8|<math>(C_2)^3:C_8</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,5)|5]] || [[SmallGroup(64,5)]] || No || || || || ||<br />
|-<br />
|64 || [[(D8:C8|6]] || [[D8:C8|<math>D_8:C_8</math>]] || No || || || || ||<br />
|-<br />
|64 || [[(Q8:C8|7]] || [[Q8:C8|<math>Q_8:C_8</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,8)|8]] || [[SmallGroup(64,8)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,9)|9]] || [[SmallGroup(64,9)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,10)|10]] || [[SmallGroup(64,10)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,11)|11]] || [[SmallGroup(64,11)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,12)|12]] || [[SmallGroup(64,12)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,13)|13]] || [[SmallGroup(64,13)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,14)|14]] || [[SmallGroup(64,14)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,15)|15]] || [[SmallGroup(64,15)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_8:C_8</math><br />
|-<br />
|64 || [[SmallGroup(64,16)|16]] || [[SmallGroup(64,16)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_8:C_8</math><br />
|-<br />
|64 || [[SmallGroup(64,17)|17]] || [[SmallGroup(64,17)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,18)|18]] || [[SmallGroup(64,18)]] || No || || || || || <math>(C_4 \times C_4):C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,19)|19]] || [[SmallGroup(64,19)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,20)|20]] || [[SmallGroup(64,20)]] || No || || || || || <math>(C_4 \times C_4):C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,21)|21]] || [[SmallGroup(64,21)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,22)|22]] || [[SmallGroup(64,22)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,23)|23]] || [[SmallGroup(64,23)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,24)|24]] || [[SmallGroup(64,24)]] || No || || || || || <math>M_{16}:C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,25)|25]] || [[SmallGroup(64,25)]] || No || || || || || <math>M_{16}:C_4</math><br />
|-<br />
|64 || [[C16xC4|26]] || [[C16xC4|<math>C_{16} \times C_4</math>]]|| <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|64 || [[SmallGroup(64,27)|27]] || [[SmallGroup(64,27)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_{16}:C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,28)|28]] || [[SmallGroup(64,28)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || <math>C_{16}:C_4</math><br />
|-<br />
|64 || [[(C2xC2):C16|29]] || [[(C2xC2):C16|<math>(C_2)^2:C_{16}</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,30)|30]] || [[SmallGroup(64,30)]] || No || || || || || <math>M_{32}:C_2</math><br />
|-<br />
|64 || [[SmallGroup(64,31)|31]] || [[SmallGroup(64,31)]] || No || || || || || <math>M_{32}:C_2</math><br />
|-<br />
|64 || [[C2wrC4|32]] || [[(C2wrC4|<math>C_2 \wr C_4</math>]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,33)|33]] || [[SmallGroup(64,33)]] || No || || || || || <br />
|-<br />
|64 || [[SmallGroup(64,34)|34]] || [[SmallGroup(64,31)]] || No || || || || || <math>(C_4 \times C_4):C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,35)|35]] || [[SmallGroup(64,35)]] || No || || || || || <math>(C_4 \times C_4):C_4</math><br />
|-<br />
|64 || [[SmallGroup(64,36)|36]] || [[SmallGroup(64,36)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,37)|37]] || [[SmallGroup(64,37)]] || No || || || || || <math>C_4:Q_8</math>, fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,38)|38]] || [[SmallGroup(64,38)]] || No || || || || || <math>D_{16}:C_4</math>, fusion trivial?<br />
|-<br />
|64 || [[SmallGroup(64,39)|39]] || [[SmallGroup(64,39)]] || No || || || || || <math>Q_{16}:C_2</math><br />
|-<br />
|64 || [[SmallGroup(64,40)|40]] || [[SmallGroup(64,40)]] || No || || || || ||<br />
|-<br />
|64 || [[SmallGroup(64,79)|79]] || [[SmallGroup(64,79)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|-<br />
|64 || [[SmallGroup(64,81)|81]] || [[SmallGroup(64,81)]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || Resistant group with automorphism group a 2-group<br />
|}<br />
--><br />
<br />
==Blocks for <math>p=3</math>==<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>1 \leq |D| \leq 27</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|- <br />
| 3 || [[C3|1]] || [[C3|<math>C_3</math>]] || 2(2) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
|9 || [[C9|1]] ||[[C9|<math>C_9</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|-<br />
|9 || [[C3xC3|2]] || [[C3xC3|<math>C_3 \times C_3</math>]] || || || || ||<br />
|-<br />
|27 || [[C27|1]] || [[C27|<math>C_{27}</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|-<br />
|27 || [[C9xC3|2]] || [[C9xC3|<math>C_9 \times C_3</math>]] || || || || ||<br />
|-<br />
|27 || [[3_+^3|3]] || [[3_+^3|<math>3_+^{1+2}</math>]] || || || || ||<br />
|-<br />
|27 || [[3_-^3|4]] || [[3_-^3|<math>3_-^{1+2}</math>]] || || || || ||<br />
|-<br />
|27 || [[C3xC3xC3|5]] || [[C3xC3xC3|<math>C_3 \times C_3 \times C_3</math>]] || || || || ||<br />
|}<br />
<br />
==Blocks for <math>p=5</math>==<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>5 \leq |D| \leq 25</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|- <br />
|5 || [[C5|1]] || [[C5|<math>C_5</math>]] ||6(6) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||<br />
|- <br />
|25 || [[C25|1]] ||[[C25|<math>C_{25}</math>]] || 6(6) || No || <math>\mathcal{O}</math> || || Max 12 classes <br />
|-<br />
|25 || [[C5xC5|2]] || [[C5xC5|<math>C_5 \times C_5</math>]] || || || || ||<br />
|}<br />
<br />
==Blocks for <math>p\geq 7</math>==<br />
<br />
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"<br />
| <strong><math>|D|</math> &nbsp;</strong><br />
|-<br />
! scope="col"| <math>|D|</math><br />
! scope="col"| SmallGroup <br />
! scope="col"| Isotype<br />
! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes<br />
! scope="col"| Complete (w.r.t.)?<br />
! scope="col"| Derived equiv classes (w.r.t)?<br />
! scope="col"| References<br />
! scope="col"| Notes<br />
|- <br />
| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || <br />
|- <br />
|7 || [[C7|1]] || [[C7|<math>C_7</math>]] ||14(14) ||No || <math>\mathcal{O}</math> || ||Max 21 classes <br />
|- <br />
|11|| [[C11|1]] || [[C11|<math>C_{11}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|- <br />
|13 || [[C13|1]] || [[C13|<math>C_{13}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|-<br />
|17|| [[C17|1]] || [[C17|<math>C_{17}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|- <br />
|19 || [[C19|1]] || [[C19|<math>C_{19}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|-<br />
|23 || [[C23|1]] || [[C23|<math>C_{23}</math>]] || ||No || <math>\mathcal{O}</math> || ||<br />
|}</div>
Charles Eaton