Endotrivial modules over groups with quaternion or semi-dihedral Sylow 2-subgroup

Abstract

Abstract. Let G be a finite group with a Sylow 2-subgroup P which is either quaternion or semi-dihedral. Let k be an algebraically closed field of characteristic 2. We prove the existence of exotic endotrivial kG-modules, whose restrictions to P are isomorphic to the direct sum of the known exotic endotrivial kP -modules and some projective modules. This provides a description of the group T (G) of endotrivial kG-modules.

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.…”

“…In particular the structure of T (G) is known when the Sylow p-subgroups of G are cyclic [26], semi-dihedral and generalized quaternion [10]. Therefore if the representation type is not wild, it only remains to treat the case of dihedral Sylow 2-subgroups.…”

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].…”

“…Specifically, this happens whenever S is not cyclic, generalized quaternion, or semi-dihedral (because, if we exclude these three cases, then T (S) is torsion-free by [12]). The excluded cases are treated in [19], [14] and [9].…”