The rest of this book is devoted to proving part (C) of the Main Theorem stated in §3.1. That is, we shall determine precisely when the members of C(G) are maximal in G, where G is any group whose socle is a classical simple group. In Chapters 6 and 7 we find all overgroups of members of C which themselves lie in C, and in Chapter 8 we find those overgroups which lie in S. The results in Chapter 8 depend on the classification of the finite simple groups, while those in Chapters 6 and 7 do not. Nevertheless, all three chapters require a fair amount of information about the simple groups, and in this chapter we survey various results from the literature concerning the finite simple groups which will be needed. The material here falls into three broad areas:

1. (1) basic properties of the simple groups (§5.1);

2. (2) subgroups of simple groups (§5.2);

3. (3) representations of the simple groups (§§5.3, 5.4).

In §5.5 we include some further results on representations of direct products of simple groups and on extensions of soluble groups by simple groups. Throughout this chapter, L will usually denote a non-abelian simple group.

Basic properties of the simple groups

As we mentioned in Chapter 1, the recent Classification Theorem asserts that the non-abelian simple groups fall into four categories: the alternating groups, the classical groups, the exceptional groups, and the sporadic groups. Of course, the alternating group An has order ½n. We display the orders of the other simple groups in Tables 5.1. A, 5.1.B and 5.1.C.

Following the classification of the finite simple groups, completed in 1980, one of the major areas of research in group theory today is the investigation of the subgroups of the finite simple groups, and in particular, the determination of their maximal subgroups. According to the classification theorem, the finite simple groups fall into four classes:

In this book we concentrate on the classical groups, which we describe in detail in Chapter 2. Our work takes as its starting point the fundamental results of M. Aschbacher in [As1]. Let G be a finite classical group. In [As1], Aschbacher introduces a large collection C(G) of natural, geometrically defined subgroups of G, and shows that almost every subgroup of G is contained in a member of C(G) (the precise result is stated in Chapter 1). Thus the collection C(G) of subgroups plays a central role in the theory of classical groups. This book is intended to be a definitive investigation of the collection C(G). In it, we solve the three main problems concerning these subgroups — namely, we determine

1. (I) the group-theoretic structure of each member of C(G),

2. (II) the conjugacy among the members of C(G),

3. (III) precisely which members of C(G) are maximal in G and which are not — and, for non-maximal members H of C(G), we determine the maximal overgroups of H in G.