The group of endotrivial modules for the symmetric and alternating groups

Abstract

We complete a classification of the groups of endotrivial modules for the modular group algebras of symmetric groups and alternating groups. We show that, for n p 2 , the torsion subgroup of the group of endotrivial modules for the symmetric groups is generated by the sign representation. The torsion subgroup is trivial for the alternating groups. The torsion-free part of the group is free abelian of rank 1 if n p 2 + p and has rank 2 if p 2 n < p 2 + p. This completes the work begun earlier by Carlson, Mazza and Nakano. = Hom B (µ, ResHowever, a direct computation shows that Res J A J A ∩B χ is not trivial. So we have a contradiction to the assumption that Y A is endotrivial.

Endotrivial modules for $p$-solvable groups

“…The most important result for our purposes is the classification of all endotrivial modules for p-groups ( [6], [7], [8], [2]). Other results have been obtained in [4], [5], [3], [15], and [16].…”

Endo-trivial modules for finite groups with Klein-four Sylow 2-subgroups

“…From the previously known cases, see e.g. [6,7,8,9,10,26], this does not seem to occur in general. Remark 1.2.…”

“…The structure of the group T (G) has been determined for some classes of general finite groups (see [5,7,26,6,9,10,24], for example), but no general solution to this problem is known.…”

The torsion group of endotrivial modules

“…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].…”