A further generalization of the Glauberman-Thompson $p$-nilpotency criterion in fusion systems

TL;DR

This paper extends classical criteria for p-nilpotency in finite groups to the setting of fusion systems, providing new conditions to determine when a fusion system is nilpotent.

Contribution

It introduces generalized criteria for p-nilpotency in fusion systems, extending the Glauberman-Thompson criterion beyond finite groups.

Findings

New criteria for nilpotency in saturated fusion systems

Extension of Glauberman-Thompson p-nilpotency criterion

Provides tools for analyzing fusion system structure

Abstract

Let $p$ be a prime and $\mathcal{F}$ be a saturated fusion system over a finite $p$-group $P$. The fusion system $\mathcal{F}$ is said to be nilpotent if $\mathcal{F}=\mathcal{F}_{P}(P)$. We provide new criteria for a saturated fusion system $\mathcal{F}$ to be nilpotent, which may be viewed as extending the Glauberman-Thompson $p$-nilpotency criterion to fusion systems.