Block library - Recent changes [en-gb]

Generic classifications by p-group class

Charles Eaton — Thu, 13 Aug 2026 16:20:40 GMT

Added [ZkZy26] results

Classification by p-group

Charles Eaton — Thu, 13 Aug 2026 15:51:32 GMT

‎Blocks for p=2: Added quaternion consequences of [EEKLS26]

References

Charles Eaton — Thu, 13 Aug 2026 15:47:42 GMT

[La16]

Status of Donovan's conjecture

Charles Eaton — Thu, 13 Aug 2026 15:14:01 GMT

Applications of [EEKLS26], Suzuki 2-groups

References

Charles Eaton — Thu, 13 Aug 2026 14:59:08 GMT

[ZqZk26], [ZkZy26], [HZZ26]

@@ Line 136: / Line 136: @@
 <math>PSU_3(q)</math> where <math>(q+1)_2=2^n</math>
-== Blocks whose defect groups are Suzuki <math>2</math>-groups ==
+== Suzuki <math>2</math>-groups ==
 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:
@@ Line 145: / Line 145: @@
 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

@@ Line 13: / Line 13: @@
 ! scope="col"| SmallGroup
 ! scope="col"| Isotype
 ! scope="col"| Known <math>k</math>-(<math>\mathcal{O}</math>-)classes
 ! scope="col"| Complete (w.r.t.)?
@@ Line 19: / Line 20: @@
 ! scope="col"| Notes
 |-
-| 1 || 1 || <math>1</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
+| 1 || 1 || <math>1</math> || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
 |-
-| 2 || [[C2|1]] || [[C2|<math>C_2</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
+| 2 || [[C2|1]] || [[C2|<math>C_2</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
 |-
-| 4 || [[C4|1]] || [[C4|<math>C_4</math>]] || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
+| 4 || [[C4|1]] || [[C4|<math>C_4</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
 |-
-| 4 || [[C2xC2|2]] || [[C2xC2|<math>C_2 \times C_2</math>]] || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Er82], [Li94] ]] ||
+| 4 || [[C2xC2|2]] || [[C2xC2|<math>C_2 \times C_2</math>]] || <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[Er82], [Li94] ]] ||
 |-
-|8 || [[C8|1]] || [[C8|<math>C_8</math>]] ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
+|8 || [[C8|1]] || [[C8|<math>C_8</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
 |-
-|8 || [[C4xC2|2]] || [[C4xC2|<math>C_4 \times C_2</math>]] ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
+|8 || [[C4xC2|2]] || [[C4xC2|<math>C_4 \times C_2</math>]] || <math>\mathcal{O}</math> ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || ||
 |-
-|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]]]
+|8 || [[D8|3]] || [[D8|<math>D_8</math>]] || <math>\mathcal{O}</math> ||6(?) || <math>k</math> || <math>k</math> || [[References|[Er87] ]] || See discussion in [[References#E|[EEKLS26]]] regarding Donovan over <math>\mathcal{O}</math>. Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]]
 |-
-|8 || [[Q8|4]] || [[Q8|<math>Q_8</math>]] ||3(3) || <math>\mathcal{O}</math> || <math>k</math> || [[References|[Er88a], [Er88b], [HKL07], [Ei16]]] ||
+|8 || [[Q8|4]] || [[Q8|<math>Q_8</math>]] || <math>\mathcal{O}</math> || 3(3) || <math>\mathcal{O}</math> || <math>k</math> || [[References|[Er88a], [Er88b], [HKL07], [Ei16]]] ||
 |-
-|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
+|8 || [[C2xC2xC2|5]] || [[C2xC2xC2|<math>C_2 \times C_2 \times C_2</math>]] || <math>\mathcal{O}</math> || 8(8) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References| [Ea16]]] || Uses CFSG
 |}
@@ Line 63: / Line 64: @@
 |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] ]] ||
 |-
-|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]]]
+|16 || [[D16|7]] || [[D16|<math>D_{16}</math>]] || <math>\mathcal{O}</math>|| 5(?) || <math>k</math> || <math>k</math> || [[References|[Er87]]] || See discussion in [[References#E|[EEKLS26]]] regarding Donovan over <math>\mathcal{O}</math>. Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]]
 |-
-|16 || [[SD16|8]] || [[SD16|<math>SD_{16}</math>]] || <math>k</math> || 7(?) || || || [[References|[Er88c], [Er90b]]] || Two other possible classes
+|16 || [[SD16|8]] || [[SD16|<math>SD_{16}</math>]] || <math>k</math> || 7(?) || || || [[References|[Er88c], [Er90b]]] || Two other possible classes. Donovan's conjecture known over <math>\mathcal{O}</math> for <math>l(B) \neq 2</math> by [[References#L|[La16]]] (see discussion in [[References#E|[EEKLS26]]]).
 |-
-|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]]]
+|16 || [[Q16|9]] || [[Q16|<math>Q_{16}</math>]] || <math>\mathcal{O}</math> || 6(?) || || <math>k</math> || [[References#E|[Er88a], [Er88b], [EEKLS26]]], [[References#H|[Ho97]]] || Classified over <math>\mathcal{O}</math> when <math>l(B)=3</math> in [[References#E|[Ei16]]]. When <math>l(B)=2</math> there are between 2 and 4 Morita equivalence classes over <math>k</math>. Finite list of Morita equivalence classes over <math>\mathcal{O}</math>, with possible repetition, by [[References#E|[EEKLS26]]]. Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20b]]]
 |-
 |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]]] ||
@@ Line 129: / Line 130: @@
 |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]]] ||
 |-
-|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]]]
+|32 || [[D32|18]] || [[D32|<math>D_{32}</math>]] || <math>\mathcal{O}</math> || 5(?) || <math>k</math> || <math>k</math> || [[References#E|[Er87]]] || See discussion in [[References#E|[EEKLS26]]] regarding Donovan over <math>\mathcal{O}</math>. Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]]
 |-
-|32 || [[SD32|19]] || [[SD32|<math>SD_{32}</math>]] || <math>k</math> || || || || ||
+|32 || [[SD32|19]] || [[SD32|<math>SD_{32}</math>]] || <math>k</math> || || || || || Donovan's conjecture known over <math>\mathcal{O}</math> for <math>l(B) \neq 2</math> by [[References#L|[La16]]] (see discussion in [[References#E|[EEKLS26]]]).
 |-
-|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]]]
+|16 || [[Q32|20]] || [[Q32|<math>Q_{32}</math>]] || <math>\mathcal{O}</math> || || || <math>k</math> || [[References#E|[Er88a], [Er88b], [EEKLS26]]], [[References#H|[Ho97]]] || Classified over <math>\mathcal{O}</math> when <math>l(B)=3</math> in [[References#E|[Ei16]]]. When <math>l(B)=2</math> there are between 2 and 4 Morita equivalence classes over <math>k</math>. Finite list of Morita equivalence classes over <math>\mathcal{O}</math>, with possible repetition, by [[References#E|[EEKLS26]]]. Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20b]]]
 |-
 |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]]]

@@ Line 176: / Line 176: @@
 |- id="L"
 |[LM80]||'''P. Landrock and G. O. Michler''', ''Principal 2-blocks of the simple groups of Ree type'', Trans. AMS '''260''' (1980), 83-111.
 |-
 |[Li94] || '''M. Linckelmann''', ''The source algebras of blocks with a Klein four defect group'', J. Algebra '''167''' (1994), 821-854.

@@ Line 21: / Line 21: @@
 |Abelian <math>3</math>-groups || No || No || [[References#K|[Ko03]]] || Puig's conjecture known for principal blocks
 |-
-|Dihedral <math>2</math>-groups || <math>k</math> || No || [[References#E|[Er87]]] ||
+|Dihedral <math>2</math>-groups || <math>\mathcal{O}</math> || No || [[References#E|[Er87]]], [[References#E|[EEKLS26]] || Full result over <math>\mathcal{O}</math> not formally stated in literature but see discussion in [[References#E|[EEKLS26]]]
 |-
 |Semidihedral <math>2</math>-groups || <math>k</math> || No || [[References#E|[Er88c], [Er90b]]] ||
@@ Line 31: / Line 31: @@
 |<math>Q_8 \times Q_8</math> || <math>\mathcal{O}</math> || No || [[References#E|[EL20]]] ||
 |-
-|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>
+|Generalised quaternion <math>2</math>-groups || <math>\mathcal{O}</math> || No || [[References#E|[Er88a], [Er88b], [Ei16], [EEKLS26]]] ||
 |-
 |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]]] ||
 |-
 |Metacyclic noncyclic <math>2</math>-groups of nonmaximal class || <math>\mathcal{O}</math> || No || [[References#C|[CG12]]], [[References#S|[Sa12b]]] || All blocks nilpotent
 |-
 |<math>p_+^{1+2}</math> for <math>p \geq 5</math> || <math>\mathcal{O}</math> || No || [[References#A|[AE23]]] ||

@@ Line 62: / Line 62: @@
 |-
 |[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]
 |-
 |[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.
@@ Line 124: / Line 126: @@
 |-
 |[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.
 |- id="J"
 |[Ja69] || '''G. Janusz''', ''Indecomposable modules for finite groups'', Ann. Math. '''89''' (1969), 209-241.
@@ Line 257: / Line 261: @@
 |[vdW91] || '''R. van der Waall''', ''On p-nilpotent forcing groups'', Indag. Mathem., N.S., '''2''' (1991), 367-384.
 |- id="W"
-|[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.
+|[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.
 |-
-|[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.
+|[ZkZy26] || '''Kun Zhang and Yuanyang Zhou''', [https://arxiv.org/abs/2604.09186 ''Inertial 2-blocks with abelian defect groups''], [https://arxiv.org/abs/2604.09186 arXiv:2604.09186]
 |}