Theorem. Let G be a finite group whose generalized Fitting subgroup is a simple classical group G o over a finite field. If 3) The representation of L on V is defined over no proper subfield of F. 4) If aeAut(F), V* is the dual of V, and V is FL-isomorphic to V *a, then either i) a = 1 and G o is orthogonal or symplectic, or ii) a is an involution and G o is unitary.The definition of cg~ appears in Sect. 1, 13 or 14, unless G o is Pf2~(q) and G contains an element inducing an automorphism of order 3 on the Dynkin diagram of Go, where no attempt is made to define a collection. Section 15 does contain some remarks relevant to this last case. The isomorphism type of the members of ~G and the action of G on ~G by conjugation is also (essentially) described in Sects. 1, 13, and t4. Presumably the members of cg G are maximal subgroups of G, with a few exceptions, but that question is not addressed here. The members of cg G are (essentially) the stabilizers of certain structures on the natural module for the classical group G o . The Main Theorem is intended to be one of the early pieces in a theory of permutation representations of finite groups based on the classification of the finite simple groups. The primitive permutation representations serve as the * Partial support supplied by the National Science Foundation

Let G = G(q) be a Chevalley group defined over a field Fq of characteristic 2. In this paper we determine the conjugacy classes of involutions in Aut(G) and the centralizers of these involutions. This study was begun in the context of a different problem.

A fusion system over a p-group S is a category whose objects form the set of all subgroups of S, whose morphisms are certain injective group homomorphisms, and which satisfies axioms first formulated by Puig that are modelled on conjugacy relations in finite groups. The definition was originally motivated by representation theory, but fusion systems also have applications to local group theory and to homotopy theory. The connection with homotopy theory arises through classifying spaces which can be associated to fusion systems and which have many of the nice properties of p-completed classifying spaces of finite groups. Beginning with a detailed exposition of the foundational material, the authors then proceed to discuss the role of fusion systems in local finite group theory, homotopy theory and modular representation theory. This book serves as a basic reference and as an introduction to the field, particularly for students and other young mathematicians.