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.