On the orbit category on nontrivial $p$-subgroups and endotrivial modules

TL;DR

This paper investigates the structure of the orbit category on nontrivial p-subgroups of finite groups and uses this to describe the group of endotrivial modules, extending previous results especially for groups with abelian or metacyclic Sylow p-subgroups.

Contribution

It determines the fundamental group of the orbit category for a broad class of finite groups and improves understanding of endotrivial modules, especially for groups with specific Sylow p-subgroup structures.

Findings

Calculated the fundamental group of the orbit category for various finite groups.

Described the group of endotrivial modules using Grodal's approach.

Extended results to groups with metacyclic Sylow p-subgroups for odd p.

Abstract

Let $p$ be a prime, let $G$ be a finite group of order divisible by $p$, and let $k$ be a field of characteristic $p$. An endotrivial $kG$-module is a finitely generated $kG$-module $M$ such that its endomorphism algebra $\operatorname{End}_kM$ decomposes as the direct sum of a one-dimensional trivial $kG$-module and a projective $kG$-module. In this article, we determine the fundamental group of the orbit category on nontrivial $p$-subgroups of $G$ for a large class of finite groups, and use Grodal's approach to describe the group of endotrivial modules for such groups. Hence, we improve on the results about the group of endotrivial modules for finite groups with abelian Sylow $p$-subgroups obtained by Carlson and Th\'evenaz. With some additional analysis, we then determine the fundamental group of the orbit category on nontrivial $p$-subgroups of $G$ and the group of endotrivial $kG$-modules in the case when $G$ has a metacyclic Sylow $p$-subgroup for $p$ odd.