Isotypies for the quasisimple groups with exceptional Schur multiplier

Let [Formula: see text] be a block with abelian defect group [Formula: see text] of a quasisimple group [Formula: see text], such that [Formula: see text] has exceptional Schur multiplier. We show that, [Formula: see text] is isotypic to its Brauer correspondent in [Formula: see text] in the sense of Broué. The proof uses methods from a previous paper [B. Sambale, Broué’s isotypy conjecture for the sporadic groups and their covers and automorphism groups, Internat. J. Algebra Comput. 25 (2015) 951–976], and relies ultimately on computer calculations. Moreover, we verify the Alperin–McKay conjecture for all blocks of [Formula: see text].

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Broué's isotypy conjecture for the sporadic groups and their covers and automorphism groups

International Journal of Algebra and Computation ◽

10.1142/s0218196715500277 ◽

2015 ◽

Vol 25 (06) ◽

pp. 951-976 ◽

Cited By ~ 3

Author(s):

Benjamin Sambale

Keyword(s):

Finite Group ◽

Simple Group ◽

Automorphism Groups ◽

Defect Group ◽

Sporadic Simple Group ◽

Sporadic Groups ◽

A Finite Group ◽

Group D

Let B be a p-block of a finite group G with abelian defect group D such that S ≤ G ≤ Aut (S), S′ = S and S/Z(S) is a sporadic simple group. We show that B is isotypic to its Brauer correspondent in N G(D) in the sense of Broué. This has been done by Rouquier for principal blocks and it remains to deal with the non-principal blocks.

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On the Structure of Integral Group Rings of Sporadic Groups

LMS Journal of Computation and Mathematics ◽

10.1112/s1461157000000309 ◽

2000 ◽

Vol 3 ◽

pp. 274-306 ◽

Cited By ~ 6

Author(s):

Frauke M. Bleher ◽

Wolfgang Kimmerle

Keyword(s):

Automorphism Groups ◽

Symmetric Groups ◽

Computational Procedure ◽

Isomorphism Problem ◽

Simple Groups ◽

Group Rings ◽

Integral Group ◽

Sporadic Groups ◽

Integral Group Rings ◽

Sporadic Simple Groups

AbstractThe object of this article is to examine a conjecture of Zassenhaus and certain variations of it for integral group rings of sporadic groups. We prove the ℚ-variation and the Sylow variation for all sporadic groups and their automorphism groups. The Zassenhaus conjecture is established for eighteen of the sporadic simple groups, and for all automorphism groups of sporadic simple groups G which are different from G. The proofs are given with the aid of the GAP computer algebra program by applying a computational procedure to the ordinary and modular character tables of the groups. It is also shown that the isomorphism problem of integral group rings has a positive answer for certain almost simple groups, in particular for the double covers of the symmetric groups.

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Finite Groups Which are Automorphism Groups of Infinite Groups Only

Canadian Mathematical Bulletin ◽

10.4153/cmb-1985-008-4 ◽

1985 ◽

Vol 28 (1) ◽

pp. 84-90

Author(s):

Jay Zimmerman

Keyword(s):

Finite Group ◽

Automorphism Group ◽

Finite Field ◽

Finite Groups ◽

Finite Simple Group ◽

Automorphism Groups ◽

Outer Automorphism ◽

Schur Multiplier ◽

Infinite Groups ◽

Infinite Set

AbstractThe object of this paper is to exhibit an infinite set of finite semisimple groups H, each of which is the automorphism group of some infinite group, but of no finite group. We begin the construction by choosing a finite simple group S whose outer automorphism group and Schur multiplier possess certain specified properties. The group H is a certain subgroup of Aut S which contains S. For example, most of the PSL's over a non-prime finite field are candidates for S, and in this case, H is generated by all of the inner, diagonal and graph automorphisms of S.

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