Preliminary definitions

It is presumed that the reader will already be familiar with the contents of this section. He is advised to read it quickly, in order to accustom himself to the notation which will be used throughout the rest of the book.

If X is a finite set, a permutation of X is a one-to-one correspondence (bisection) α : X → X. Two such permutations α, β can be composed to give the permutation αβ : X → X, which we shall define by the rule αβ(x) = α(β(x)). That is, we shall write functions on the left, and compose in the order compatible with this convention. Under the operation of composition the set of all permutations of X forms a group Sym(X), the symmetric group on X. If X is the set {1, 2, …, n}, we write Sn for Sym(X), and we have |sn | = n! where we use | to denote cardinality.

If G is a subgroup of Sym(X), then we shall say that the pair (G, X) is a permutation group of degree | x |, and that G acts on X. More generally, we shall meet the situation where there is a homomorphism g ⇝ ĝ of a group G into Sym(X): this will be called a permutation representation of G.

This book replaces an earlier one, entitled ‘Finite Groups of Automorphisms’ (No. 6 in the L. M. S. Lecture Note Series). When it became clear that the original book was still in demand, it was decided that a complete revision was preferable to a reprint. In this way we hoped to incorporate the fruits of experience, gleaned from readers' comments, and at the same time, to keep the book up-to-date by introducing some new material.

The entire book has been rewritten. In some sections of Chapters 1, 2 and 3 the earlier version has been followed quite closely, while in others (notably 1. 5, 1. 6, 2. 4, 2. 5, 3. 4, 3. 5), the material is treated differently. Chapter 4 (Groups and Graphs) is presented from a new viewpoint, beginning with the graphical representation of permutation groups. The algebraic theory of the adjacency matrix has been restricted to the case of strongly regular graphs, since the generalization to graphs of larger diameter is discussed elsewhere (reference [1] of Chapter 4). The material on the Higman-Sims group has been amplified. Chapter 5 (Maps) is completely new. We hope that its inclusion will lead to renewed interest in a subject where ideas from many different areas of mathematics come together.

We have tried to make the book suitable for use as a course text at advanced undergraduate or postgraduate level. For this reason, there are thirteen ‘project’ sections, which should provide a good test of a student's real understanding of the text. The topics treated in the project sections are occasionally needed in subsequent chapters. (See also the note below.)

Maps and surfaces

In this chapter we shall discuss graphs realized by a set of points and lines on a closed orientable surface. Although this notion arises in a topological context, we shall show that it is possible to develop it by purely combinatorial arguments. The reader who is unfamiliar with the topology of surfaces should not be at a disadvantage.

We begin at an intuitive level. For our purposes, it is sufficient to say that a ‘surface’ is a compact topological space which has two special properties:

1. (i) it is locally homeomorphic to ordinary Euclidean 2-space;

2. (ii) it has a consistent global orientation.

The sphere and the torus are the simplest examples. The Euclidean plane is not compact, and so it is not a ‘surface’ for us; however, it may be made homeomorphic to a sphere by the addition of a single point, and so the two spaces have very similar properties. The Klein bottle is not allowed, since it has no consistent global orientation.

Let us suppose that a graph Γ is represented by a set of points and lines on a surface, in such a way that the lines intersect only at the points representing their end vertices.