Introduction

Control of fusion

Let G be a finite group, and let p be a prime number.

Definition 1.1. We say that a subgroup H of G controls the fusion of p-subgroups of G if the following two conditions are fulfilled:

(C1) H contains a Sylow p-subgroup Sp of G,

(C2) whenever P is a subgroup of Sp and g is an element of G such that gPg-1 ⊆ Sp, there exist z in the centralizer CG(P) of P in G, and h in H, such that g = hz.

Example 1.2. (The basic example) We denote by Op′(G) the largest normal subgroup of G with order prime to p. Then if H is a subgroup of G which “covers the quotient” G/OP′(G) (i.e., if G = HOP′(G)), then H controls the fusion of p-subgroups of G.

The following two results provide fundamental examples where the converse is true. The first one is due to Frobenius and was proved in 1905. The second one was proved by Glauberman for the case p = 2 (see [Gl]), and for odd p it is a consequence of the classification of non abelian finite simple groups (see also [Ro] for an approach not using the classification).