We begin with a brief review of the standard ways in which finite groups of Lie type are classified, constructed, and described. One complication is the multiplicity of approaches to describing this family of groups, which leads in turn to differing notational conventions in the literature. Our viewpoint will be mainly that of algebraic groups over finite fields (1.1), reformulated in terms of Frobenius maps (1.3). But occasional use will be made of the convenient axiomatic approach afforded by BN-pairs: see 1.7 and Chapter 7 below.

Even within the framework of algebraic groups, there is more than one way to organize the finite groups. Steinberg's unified description of the groups as fixed points of endomorphisms of algebraic groups is undoubtedly the most elegant and useful. However, in our treatment of modular representations it will be convenient to keep the groups of Ree and Suzuki (defined only in characteristic 2 or 3) largely separate from the other groups: see Chapter 20. These groups arise less directly from the ambient algebraic groups and of course do not exhibit any “generic” behavior for large primes p as other groups of Lie type do.

To conclude this introductory chapter we establish in 1.8 some standard notation.

Algebraic Groups over Finite Fields

The finite groups of Lie type are close relatives of the groups G(k) of rational points of algebraic groups defined over a finite field k.