Endotrivial modules for the sporadic simple groups and their covers

Abstract

Abstract. In a step towards the classification of endotrivial modules for quasi-simple groups, we investigate endotrivial modules for the sporadic simple groups and their covers. A main outcome of our study is the existence of torsion endotrivial modules with dimension greater than one for several sporadic groups with p-rank greater than one.

The torsion group of endotrivial modules

“…Several examples are given in [17] for various sporadic groups. The purpose of this section is to provide two explicit examples for classical groups with an abelian Sylow p-subgroup.…”

“…The computing time depends on such things as the size of the permutation representation of G and the number of subgroups of S. Here we list only groups where K(G) is not trivial. The results should be compared with those of [15,16,17]. The notation for the groups is the Atlas notation.…”

“…They are modules which have universal deformation rings [13]. The endotrivial modules have been classified in case G is a p-group ( [11,12]) and various results have appeared since for some specific families of groups [4,5,6,7,8,9,10,18,19,20,17]. Recently, another line of research has developed that is concerned with the classification of all endotrivial modules which are simple [21,15,16].…”

Trivial-source endotrivial modules for sporadic groups

“…The determination of T (G, S) is the main problem in endotrivial modules for arbitrary finite groups. Lassueur and Mazza (2015b) made some headway on the problem of determining T (G, S) for G a quasisimple sporadic group by computing it for small sporadic groups (the Monster was considered in Grodal (2016)), and Carlson et al (2009) and Carlson et al (2010) determine T (G, S) for G an alternating or symmetric group (their covering groups were studied in Lassueur and Mazza (2015a)). Both results contain minor errors (T (G, S) for G = 3.J 3 and p = 2 should be 1, not 3, and for G = A 2 p , A 2 p+1 and p > 3 it should be 4, not 2 × 2), the former corrected here and the latter in Grodal (2016).…”

Endotrivial modules : a reduction to p′-central extensions

“…Endo-trivial modules play an important role in the representation theory of finite groups in prime characteristic p. They have been classified in a number of special cases (see the recent papers [CMN15,LM15b] and the references therein). Over an algebraically closed field k of prime characteristic p, endo-trivial modules for a finite group G form an abelian group T (G), which is known to be finitely generated.…”