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

TL;DR

This paper classifies endotrivial modules for groups between SL(n,q) and GL(n,q) over algebraically closed fields of characteristic p, completing the classification for all finite groups of Lie Type A.

Contribution

It determines the group of endotrivial modules for these groups, extending previous work to cover all finite groups of Lie Type A.

Findings

Complete classification of endotrivial modules for groups of Lie Type A.

Determination of the group T(G/Z) for specified groups.

Results applicable to algebraically closed fields of characteristic p not dividing q.

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.