2 - Finite classical groups

Summary

In this chapter we provide a brief introduction to the finite classical groups. Our treatment is tailored towards the application to derangements explained in Chapter 1, and we focus on forms, standard bases and automorphisms. We also describe some specific classical group embeddings (or constructions, as we call them), which will be useful later. Finally, in Section 2.6 we briefly discuss Aschbacher's theorem [3] on the subgroup structure of finite classical groups, which plays an essential role in our analysis of derangements in primitive almost simple classical groups.

There are a number of excellent general references on finite classical groups, which provide a more comprehensive treatment. In particular, we refer the reader to Dieudonné [47] and Taylor [118]. Other good references include Chapter 2 in [86], Chapter 1 in [13], Chapter 3 in [122], Chapter 7 in [4], and Cameron's lecture notes on classical groups [36]. Sections 27 and 28 in the recent book by Malle and Testerman [99] provide an accessible account of Aschbacher's subgroup structure theorem (also see Chapter 2 in [13]).

Linear groups

Let V be an n-dimensional vector space over the finite field F_q with q= p^f for a prime number p. The general linear group GL(V) is the group of all invertible linear transformations of V. The centre Z of GL(V) is the subgroup of all scalar transformations v ⟼ λ v with λ ∈ F_q^×. The group GL(V) acts naturally on the set of all 1-dimensional subspaces of V, and the kernel of this action is Z. We define PGL(V) = GL(V)/Z, the projective general linear group, which is isomorphic to the permutation group induced by GL(V) in this action.

By fixing a basis for V we can represent each element of GL(V) by an invertible n×n matrix with entries in F_q. We denote the group of all such matrices by GL_n(q). In this representation of GL(V), the centre Z consists of all scalar matrices λI_n, where I_n is the n×n identity matrix and λ ∈ F_q^× is a nonzero scalar. We denote GL_n(q)/Z by PGL_n(q).