  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.
