Introduction

This text is about the interaction between algebra and geometry, and central to this interaction is the idea of a group. Groups are studied as abstract systems in algebra; they help us to describe the arithmetic structure of the real and complex numbers, and modular arithmetic, and they provide a framework for a discussion of permutations of an arbitrary set. Groups also arise naturally in geometry; for example, as the set of translations of the plane, the rotations of the plane about the origin, the symmetries of a cube, and the set of all functions of the plane into itself that preserve distance. We shall see that geometry provides many other interesting examples of groups and, in return, group theory provides a language and a number of fundamental ideas which can be used to give a precise description of geometry. In 1872 Felix Klein proposed his Erlangen Programme in which, roughly speaking, he suggested that we should study different geometries by studying the groups of transformations acting on the geometry. It is this spirit that this text has tried to capture.

We shall assume familiarity with the most basic facts about elementary set theory. We recall that if X is any set, then x ∈ X means that x is an element, or member, of X, and x ∉ X means that x is not an element of X.