MediaWiki API result
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"19": {
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"title": "Reductions",
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"*": "This page will contain descriptions of reduction techniques and results.\n\n== Donovan's conjecture ==\n\n[[Statements of conjectures#Donovan's conjecture|For the statement of the conjecture click here.]]\n\n=== <math>k</math>-Donovan conjecture ===\n\nBy [[References#K|[K\u00fc95]]] 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.\n\nSeveral reductions were achieved in [[References#D|[Du04]]], but these have been subsumed in later work.\n\n'''<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]]].\n\n\n=== <math>\\mathcal{O}</math>-Donovan conjecture ===\n\nEisele in [[References#E|[Ei18]]] proved the analogue of [[References#K|[K\u00fc95]]] for the <math>\\mathcal{O}</math>-Donovan conjecture, so it suffices to consider blocks of finite groups that are generated by the defect groups.\n\nBy [[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>. \n\n'''<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]]].\n\n'''<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]]].\n\n== Weak Donovan conjecture ==\n\nFor 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]]]."
}
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"11": {
"pageid": 11,
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"title": "References",
"revisions": [
{
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"*": "[[#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,]] \n\n{| \n|- id=\"A\"\n|[Al79] || '''J. L. Alperin''', ''Projective modules for <math>SL(2,2^n)</math>'', J. Pure and Applied Algebra '''15''' (1979), 219-234.\n|-\n|[Al80] || '''J. L. Alperin''', ''Local representation theory'', The Santa Cruz Conference on Finite Groups., Proc. Sympos. Pure Math. '''37''' (1980), 369-375.\n|-\n|[AE81] || '''J. L. Alperin and L. Evens''', ''Representations, resoluutions and Quillen's dimension theorem'', J. Pure Appl. Algebra '''22''' (1981), 1-9.\n|-\n|[AE04] || '''Jianbei An and C. W. Eaton''', ''Blocks with trivial intersection defect groups'', Math. Z. '''247''' (2004), 461-486.\n|-\n|[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]\n|-\n|[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.\n|-\n|[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]\n|-\n|[AS20] || '''C. G. Ardito and B. Sambale''', [http://www.advgrouptheory.com/journal/Volumes/12/ArditoSambale.pdf ''Brou\u00e9's Conjecture for 2-blocks with elementary abelian defect groups of order 32''], Advances in Group Theory and Applications 12 (2021), 71\u201378. \n|-\n|[AKO11] || '''M. Aschbacher, R. Kessar and B. Oliver''', ''Fusion systems in algebra and topology'', London Mathematical Society Lecture Notes '''391''', Cambridge University Press (2011).\n|- id=\"B\"\n|[BK07] || '''D. Benson and R. Kessar''', ''Blocks inequivalent to their Frobenius twists'', J. Algebra '''315''' (2007), 588-599.\n|-\n|[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]\n|-\n|[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.\n|-\n|[Bra41] || '''R. Brauer''', ''Investigations on group characters'', Ann. Math. '''42''' (1941), 936-958.\n|-\n|[BP80] || '''M. Brou\u00e9 and L. Puig''', ''A Frobenius theorem for blocks'', Invent. Math. '''56''' (1980), 117-128.\n|-\n|[BP80b] || '''M. Brou\u00e9 and L. Puig''', ''Characters and local structure in G-algebras'', J. Algebra '''63''' (1980), 306-317.\n|- id=\"C\"\n|[Cr11] || '''D. A. Craven''', ''The Theory of Fusion Systems: An Algebraic Approach'', Cambridge University Press (2011).\n|-\n|[Cr12] || '''D. A. Craven''', [https://arxiv.org/abs/1207.0116 ''Perverse Equivalences and Brou\u00e9's Conjecture II: The Cyclic Case''], [https://arxiv.org/abs/1207.0116 arXiv:1207.0116]\n|-\n|[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] \n|-\n|[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.\n|-\n|[CG12] || '''D. A. Craven and A. Glesser''', ''Fusion systems on small p-groups'', Trans. AMS '''364''' (2012) 5945-5967.\n|-\n|[CR13] || '''D. A. Craven and R. Rouquier''', ''Perverse equivalences and Brou\u00e9's conjecture'', Adv. Math. '''248''' (2013), 1-58.\n|-\n|[CuRe81a] || '''C. W. Curtis and I. Reiner''', ''Methods of representation theory, with applications to finite groups and orders, Volume I'', Wiley-Interscience (1981).\n|-\n|[CuRe81b] || '''C. W. Curtis and I. Reiner''', ''Methods of representation theory, with applications to finite groups and orders, Volume II'', Wiley-Interscience (1981).\n|- id=\"D\"\n|[Da66] || '''E. C. Dade''', ''Blocks with cyclic defect groups'', Ann. Math. '''84''' (1966), 20-48. \n|-\n|[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]\n|-\n|[Du14] || '''O. Dudas''', [https://arxiv.org/abs/1011.5478 ''Coxeter orbits and Brauer trees II''], Int. Math. Res. Not. '''15''' (2014), 4100-4123.\n|-\n|[D\u00fc04] || '''O. D\u00fcvel''', ''On Donovan's conjecture'', J. Algebra '''272''' (2004), 1-26.\n|- id=\"E\"\n|[Ea16] || '''C. W. Eaton''', ''Morita equivalence classes of <math>2</math>-blocks of defect three'', Proc. AMS '''144''' (2016), 1961-1970.\n|-\n|[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]\n|-\n|[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]\n|-\n|[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.\n|-\n|[EKKS14] || '''C. W. Eaton, R. Kessar, B. K\u00fclshammer and B. Sambale''', ''<math>2</math>-blocks with abelian defect groups'', Adv. Math. '''254''' (2014), 706-735.\n|-\n|[EKS12] || '''C. W. Eaton, B. K\u00fclshammer and B. Sambale''', ''<math>2</math>-blocks with minimal nonabelian defect groups, II'', J. Group Theory '''15''' (2012), 311-321.\n|-\n|[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.\n|-\n|[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.\n|-\n|[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.\n|-\n|[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.\n|-\n|[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\n|-\n|[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.\n|-\n|[Ei18] || '''F. Eisele''', ''[https://arxiv.org/abs/1807.05110 The Picard group of an order and K\u00fclshammer reduction]'', Algebr. Represent. Theory '''24''' (2021), 505-518.\n|-\n|[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. \n|-\n|[EiLiv20] || '''F. Eisele and M. Livesey''', ''[https://arxiv.org/abs/2006.13837 Arbitrarily large Morita Frobenius numbers]'', Algebra Number Theory '''16''' (2022), 1889-1904.\n|-\n|[Er82] || '''K. Erdmann''', ''Blocks whose defect groups are Klein four groups: a correction'', J. Algebra '''76''' (1982), 505-518.\n|-\n|[Er87] || '''K. Erdmann''', ''Algebras and dihedral defect groups'', Proc. LMS '''54''' (1987), 88-114.\n|-\n|[Er88a] || '''K. Erdmann''', ''Algebras and quaternion defect groups, I'', Math. Ann. '''281''' (1988), 545-560.\n|-\n|[Er88b] || '''K. Erdmann''', ''Algebras and quaternion defect groups, II'', Math. Ann. '''281''' (1988), 561-582. \n|-\n|[Er88c] || '''K. Erdmann''', ''Algebras and semidihedral defect groups I'', Proc. LMS '''57''' (1988), 109-150. \n|-\n|[Er90] || ''' K. Erdmann''', ''Blocks of tame representation type and related algebras'', Lecture Notes in Mathematics '''1428''', Springer-Verlag (1990).\n|-\n|[Er90b] || '''K. Erdmann''', ''Algebras and semidihedral defect groups II'', Proc. LMS '''60''' (1990), 123-165.\n|- id=\"F\"\n|[Fa17] || '''N. Farrell''', ''On the Morita Frobenius numbers of blocks of finite reductive groups'', J. Algebra '''471''' (2017), 299-318.\n|-\n|[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.\n|- id=\"G\"\n|[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\u2013274.\n|-\n|[GO97] || '''H. Gollan and T. Okuyama''', ''Derived equivalences for the smallest Janko group'', preprint (1997).\n|-\n|[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\n|- id=\"H\"\n|[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.\n|-\n|[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.\n|-\n|[Hi63] || '''G. Higman''', ''Suzuki 2-groups'', Illinois J. Math. '''7''' (1963), 79\u201396.\n|- \n|[HK00] || '''G. Hiss and R. Kessar''', ''Scopes reduction and Morita equivalence classes of blocks in finite classical groups'', J. Algebra '''230''' (2000), 378-423.\n|-\n|[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.\n|-\n|[Ho97] || '''T. Holm''', ''Derived equivalent tame blocks'', J. Algebra '''194''' (1997), 178-200.\n|-\n|[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\u201343.\n|- id=\"J\"\n|[Ja69] || '''G. Janusz''', ''Indecomposable modules for finite groups'', Ann. Math. '''89''' (1969), 209-241.\n|-\n|[Jo96] || '''T. Jost''', ''Morita equivalences for blocks of finite general linear groups'', Manuscripta Math. '''91''' (1996), 121-144.\n|- id=\"K\"\n|[Ke96] || '''R. Kessar''', ''Blocks and source algebras for the double covers of the symmetric and alternating groups'', J. Algebra '''186''' (1996), 872-933.\n|-\n|[Ke00] || '''R. Kessar''', ''Equivalences for blocks of the Weyl groups'', Proc. Amer. Math. Soc. '''128''' (2000), 337-346.\n|-\n|[Ke01] || '''R. Kessar''', ''Source algebra equivalences for blocks of finite general linear groups over a fixed field'', Manuscripta Math. '''104''' (2001), 145-162. \n|-\n|[Ke02] || '''R. Kessar''', ''Scopes reduction for blocks of finite alternating groups'', Quart. J. Math. '''53''' (2002), 443-454.\n|-\n|[Ke05] || ''' R. Kessar''', ''A remark on Donovan's conjecture'', Arch. Math (Basel) '''82''' (2005), 391-394.\n|-\n|[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]\n|-\n|[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\u201358.\n|-\n|[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.\n|-\n|[KKW02] || '''S. Koshitani, N. Kunugi and K. Waki''', ''Brou\u00e9's conjecture for non-principal 3-blocks of finite groups'', J. Pure and Applied Algebra '''173''' (2002), 177-211. \n|-\n|[KKW04] || '''S. Koshitani, N. Kunugi and K. Waki''', ''Brou\u00e9's abelian defect group conjecture for Held group and the sporadic Suzuki group'', J. Algebra '''279''' (2004), 638-666. \n|-\n|[KoLa20] || '''S. Koshitani and C. Lassueur''', ''Splendid Morita equivalences for principal 2-blocks with dihedral defect groups'', Math. Z. '''294''' (2020), 639-666.\n|-\n|[KoLa20b] || '''S. Koshitani and C. Lassueur''', ''Splendid Morita equivalences for principal blocks with generalised quaternion defect groups'', J. Algebra '''558''' (2020), 523-533.\n|-\n|[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.\n|-\n|[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]\n|-\n|[K\u00fc80] || '''B. K\u00fclshammer''', ''On 2-blocks with wreathed defect groups'', J. Algebra '''64''' (1980), 529\u2013555.\n|-\n|[K\u00fc81] || '''B. K\u00fclshammer''', ''On p-blocks of p-solvable groups'', Comm. Alg. '''9''' (1981), 1763-1785.\n|-\n|[K\u00fc91] || '''B. K\u00fclshammer''', ''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).\n|-\n|[K\u00fc95] || '''B. K\u00fclshammer''', ''Donovan's conjecture, crossed products and algebraic group actions'', Israel J. Math. '''92''' (1995), 295-306.\n|-\n|[KS13] || '''B. K\u00fclshammer and B. Sambale''', ''The 2-blocks of defect 4'', Representation Theory '''17''' (2013), 226-236.\n|-\n|[Ku00] || '''N. Kunugi''', ''Morita equivalent 3-blocks of the 3-dimensional projective special linear groups'', Proc. LMS '''80''' (2000), 575-589.\n|-\n|[Kup69] || '''H. Kupisch''', ''Unzerlegbare Moduln endlicher Gruppen mit zyklischer p-Sylow Gruppe'', Math. Z. '''108''' (1969), 77-104.\n|- id=\"L\"\n|[LM80]||'''P. Landrock and G. O. Michler''', ''Principal 2-blocks of the simple groups of Ree type'', Trans. AMS '''260''' (1980), 83-111.\n|-\n|[Li94] || '''M. Linckelmann''', ''The source algebras of blocks with a Klein four defect group'', J. Algebra '''167''' (1994), 821-854.\n|-\n|[Li94b] || '''M. Linckelmann''', ''A derived equivalence for blocks with dihedral defect groups'', J. Algebra '''164''' (1994), 244-255. \n|-\n|[Li96] || '''M. Linckelmann''', ''The isomorphism problem for cyclic blocks and their source algebras'', Invent. Math. '''125''' (1996), 265-283.\n|-\n|[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]\n|-\n|[Li18b] || '''M. Linckelmann''', ''Finite-dimensional algebras arising as blocks of \ufb01nite group algebras'', Contemporary Mathematics '''705''' (2018), 155-188.\n|-\n|[Li18c] || '''M. Linckelmann''', ''The block theory of finite group algebras, Volume 1'', London Math. Soc. Student Texts '''92''', Cambridge University Press (2018).\n|-\n|[Li18d] || '''M. Linckelmann''', ''The block theory of finite group algebras, Volume 2'', London Math. Soc. Student Texts '''92''', Cambridge University Press (2018).\n|-\n|[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\u20131088\n|-\n|[Liv19] || '''M. Livesey''', [https://arxiv.org/abs/1907.12167 ''On Picard groups of blocks with normal defect groups''], J. Algebra '''566''' (2021), 94-118.\n|-\n|[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]\n|-\n|[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]\n|- id=\"M\"\n|[Mac] || '''N. Macgregor''', ''Morita equivalence classes of tame blocks of finite groups'', J. Algebra '''608''' (2022), 719-754.\n|-\n|[Mar] || '''C. Marchi''', ''Picard groups for blocks'', PhD thesis, University of Manchester (2022)\n|-\n|[Ma86] || '''U. Martin''', ''Almost all <math>p</math>-groups have automorphism group a <math>p</math>-group'', Bull. AMS '''15''' (1986), 78-82.\n|-\n|[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\u201348.\n|-\n|[MS08] || '''J. M\u00fcller and M. Schaps''', ''The Brou\u00e9 conjecture for the faithful 3-blocks of <math>4.M_{22}</math>'', J. 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