On the Structure of Integral Group Rings of Sporadic Groups

Abstract

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.