# On p-stability in groups and fusion systems

## TL;DR

This paper extends the concept of p-stability from finite groups to fusion systems, analyzing the involvement of Qd(p) in simple groups and establishing new fusion-theoretic stability properties and theorems.

## Contribution

It introduces p-stability for fusion systems, characterizes its properties, and proves a fusion-theoretic version of Thomson's maximal subgroup theorem and Glauberman's theorem.

## Key findings

- Qd(p) involvement in simple groups leads to specific subgroup structures
- Fusion systems exhibit p-stability with new characterizations
- Fusion-theoretic theorems analogous to classical group results are established

## Abstract

The aim of this paper is to generalise the notion of p-stability to fusion systems. We study the question how Qd(p) is involved in finite simple groups. We show that with a single exception a simple group involving Qd(p) has a subgroup isomorphic to either Qd(p) or a central extension of it by a cyclic group of order p. We define p-stability for fusion systems, characterise some of its properties and prove a fusion theoretic version of Thomson's maximal subgroup theorem. We introduce the notion of section p-stability both for groups and fusion systems and prove a version of Glauberman's theorem to fusion systems.