Introduction to Groups

Abstract

A theory of groups first began to take form at the end of the eighteenth century. It developed slowly and attracted very little notice during the first decades of the nineteenth century. Then, in a few years centering about 1830, the theory of groups took a giant leap forward and made a major contribution to the general development of mathematics in the work of Galois and Abel on the solvability of algebraic equations. Since then, the concepts underlying the theory of groups have been elaborated and extended into many branches of mathematics. There have been applications to such diverse fields as quantum mechanics, crystallography, and the theory of knots. This book is concerned with groups and their graphical representation. Our first task is to clarify what is meant by a group. One basic idea that reaches to the very essence of the group concept is the notion of structure, or pattern. In what follows, the reader will see the unfolding of a succession of examples and explanations, definitions and theorems, all calculated to be variations on one fundamental theme: how groups and their graphs embody and illustrate one kind of mathematical structure. So far, we have been using the word “group” without giving the reader any idea of what the word means. To present a complete formal definition at one fell swoop might leave the reader as mystified as he was to start with. We shall therefore develop the concept of a group gradually, and we begin by presenting two examples.