1 - Preliminary Results

Summary

We take as our starting point the text Finite Group Theory [FGT], although we need only a fraction of the material in that text. Frequently quoted results from [FGT] will be recorded in this chapter and in other of the introductory chapters.

Chapters 1 and 2 record some of the most basic terminology and notation we will be using plus some elementary results. The reader should consult [FGT] for other basic group theoretic terminology and notation, although we will try to recall such notation when it is first used, or at least give a specific reference to [FGT] at that point. There is a “List of Symbols” at the end of [FGT] which can be used to help hunt down notation.

We begin in Section 1 with a brief discussion of abstract representations of groups. Then in Section 2 we specialize to permutation representations. In Section 3 we consider graphs and in Section 4 geometries (in the sense of J. Tits) and geometric complexes. In the last few sections of the chapter we record a few basic facts about the general linear group and fiber products of groups.

Abstract representations

Let C be a category. For χ an object in C, we write Aut(χ) for the group of automorphisms of χ under the operation of composition in C (cf. Section 2 in [FGT]). A representation of a group G in the category C is a group homomorphism π; G → Aut(χ). For example, a permutation representation is a representation in the category of sets and a linear representation is a representation in the category of vector spaces and linear maps.