The torsion group of endotrivial modules

TL;DR

This paper classifies torsion endotrivial modules for finite groups with abelian Sylow p-subgroups by analyzing the kernel of the restriction map from the group of endotrivial modules of G to that of S.

Contribution

It provides a structural classification of torsion endotrivial modules for finite groups with abelian Sylow p-subgroups based on the kernel of the restriction map.

Findings

Determined the kernel of the restriction map in terms of group structure.

Classified all torsion endotrivial modules for groups with abelian Sylow p-subgroups.

Abstract

Let G be a finite group and let T(G) be the abelian group of equivalence classes of endotrivial kG-modules, where k is an algebraically closed field of characteristic p. We determine, in terms of the structure of G, the kernel of the restriction map from T(G) to T(S), where S is a Sylow p-subgroup of G, in the case when S is abelian. This provides a classification of all torsion endotrivial kG-modules in that case.