Introduction

Let H be a finite group and let p be a prime number dividing |H|. For S ∈ SylpH, put Loc(S) = {NH(R) | 1 ≠ R ≤ S}. A p-local subgroup of H is a subgroup of H which is in Loc(S) for some S ∈ SylpH.

The modern era of p-local analysis (meaning the study of various collections of p-local subgroups) may be considered to have its origin in Thompson's thesis [T1] (see also [T2] and [T3]). There he introduced what is now known as the Thompson order and subsequent refinements of this idea were important in the proof of the Odd Order Theorem [FT]. Questions concerning p-local subgroups frequently play a role in important problems about finite groups. This has been particularly true of the work which resulted in the classification of the finite simple groups.

In 1980 Goldschmidt inaugurated the amalgam method when in [Gol] he used amalgams to study a configuration arising in the N-group paper. More recently the amalgam method has been deployed in the revision of the simple group classification (see, for example, [S1]).

The basic situation in which the amalgam method applies is given in

Hypothesis 1.1.G is a group containing distinct finite subgroups P1 and P2 such that

1. (i) G = 〈P1, P2〉; and

2. (ii) P1 ∪ P2 contains no non-trivial normal subgroups of G.