Endotrivial Modules for Finite Groups of Lie Type A in Nondefining Characteristic

Abstract

Let $G$ be a finite group such that $\text{SL}(n,q)\subseteq G\subseteq \text{GL}(n,q)$ and $Z$ be a central subgroup of $G$. In this paper we determine the group $T(G/Z)$ consisting of the equivalence classes of endotrivial $k(G/Z)$-modules where $k$ is an algebraically closed field of characteristic $p$ such that $p$ does not divide $q$. The results in this paper complete the classification of endotrivial modules for all finite groups of Lie Type $A$, initiated earlier by the authors.