On Endo-trivial Modules for p-Solvable Groups

TL;DR

This paper proves that for certain p-nilpotent groups with specific subgroups, all simple endo-trivial modules over an algebraically closed field are one-dimensional, confirming a conjecture in modular representation theory.

Contribution

It establishes a proof of a conjecture regarding the structure of simple endo-trivial modules for p-nilpotent groups with elementary Abelian p-subgroups.

Findings

All simple endo-trivial modules are 1-dimensional for the specified groups.

The result confirms the conjecture of Carlson, Mazza, and Thévenaz.

The proof applies to groups containing non-cyclic elementary Abelian p-subgroups.

Abstract

We prove a conjecture of J. Carlson, N. Mazza and J. Th\'evenaz; namely, we will prove that if $G$ is a finite $p$-nilpotent group which contains a non-cyclic elementary Abelian $p$-subgroup and $k$ is an algebraically closed field of characteristic $p$, then all simple endo-trivial $kG$-modules are $1$-dimensional.