On p-stability in groups and fusion systems

The aim of this paper is to generalise the notion of p-stability (p is an odd prime) in finite group theory to fusion systems. We first compare the different definitions of p-stability for groups and examine properties of p -stability concerning subgroups and factor groups. Motivated by Glauberman's theorem, we study the question of 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 Qd(p) by a cyclic group of order p. Then we define p-stability for fusion systems and characterise some of its properties. We prove a fusion theoretic version of Thompson'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. We also examine relationship between solubility and p -stability for fusion systems and determine the simple groups whose fusion systems are Qd(p)-free.