On twisted conjugacy classes of type D in sporadic simple groups

Let [Formula: see text] denote the group whose Cayley graph with respect to a particular generating set is the Diestel–Leader graph [Formula: see text], as described by Bartholdi, Neuhauser and Woess. We compute both [Formula: see text] and [Formula: see text] for [Formula: see text], and apply our results to count twisted conjugacy classes in these groups when [Formula: see text]. Specifically, we show that when [Formula: see text], the groups [Formula: see text] have property [Formula: see text], that is, every automorphism has an infinite number of twisted conjugacy classes. In contrast, when [Formula: see text] the lamplighter groups [Formula: see text] have property [Formula: see text] if and only if [Formula: see text].

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On conjugacy classes of maximal subgroups of finite simple groups, and a related zeta function

Duke Mathematical Journal ◽

10.1215/s0012-7094-04-12834-9 ◽

2005 ◽

Vol 128 (3) ◽

pp. 541-557 ◽

Cited By ~ 14

Author(s):

Martin W. Liebeck ◽

Benjamin M. S. Martin ◽

Aner Shalev

Keyword(s):

Zeta Function ◽

Conjugacy Classes ◽

Maximal Subgroups ◽

Simple Groups ◽

Finite Simple Groups

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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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