A characteristic subgroup for fusion systems

TL;DR

This paper generalizes Stellmacher's characteristic subgroup construction from finite groups to fusion systems, providing a new tool for analyzing the structure of fusion systems related to prime 2 and odd primes.

Contribution

It extends Stellmacher's subgroup W(S) concept to fusion systems, introducing a Glauberman functor applicable to odd primes.

Findings

W(S) is normal in certain fusion systems

The construction applies to odd primes

Provides a new perspective on fusion system structure

Abstract

As a counterpart for the prime 2 to Glauberman's $ZJ$-theorem, Stellmacher proves that any nontrivial 2-group $S$ has a nontrivial characteristic subgroup $W(S)$ with the following property. For any finite $\Sigma_4$-free group $G$, with $S$ a Sylow 2-subgroup of $G$ and with $O_2(G)$ self-centralizing, the subgroup $W(S)$ is normal in $G$. We generalize Stellmacher's result to fusion systems. A similar construction of $W(S)$ can be done for odd primes and gives rise to a Glauberman functor.