The torsion group of endotrivial modules

Abstract

Let $G$ be a finite group and let $T\left(G\right)$ 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 $T\left(G\right)$ to $T\left(S\right)$, 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.