Endotrivial modules for the general linear group in a nondefining characteristic

Abstract

Suppose that $$G$$ is a finite group such that $$\mathrm{SL }(n,q)\subseteq G \subseteq \mathrm{GL }(n,q)$$ , and that $$Z$$ is a central subgroup of $$G$$ . Let $$T(G/Z)$$ be the abelian group of equivalence classes of endotrivial $$k(G/Z)$$ -modules, where $$k$$ is an algebraically closed field of characteristic $$p$$ not dividing $$q$$ . We show that the torsion free rank of $$T(G/Z)$$ is at most one, and we determine $$T(G/Z)$$ in the case that the Sylow $$p$$ -subgroup of $$G$$ is abelian and nontrivial. The proofs for the torsion subgroup of $$T(G/Z)$$ use the theory of Young modules for $$\mathrm{GL }(n,q)$$ and a new method due to Balmer for computing the kernel of restrictions in the group of endotrivial modules.