Chapter 1

Introduction

Here we introduce some of the topics in this book—briefly, but we

hope invitingly. The ideas will be developed more fully and their

inter-relations more thoroughly examined in the chapters that follow.

This book has two over-arching themes. One is that different

parts of mathematics can and do come together in surprising and

illuminating ways: suggesting questions, providing tools, and gener-

ating examples. The other is the idea of a difference set—a special

subset of a group. It exemplifies the first theme, since it belongs both

to group theory and to combinatorics, and the study of difference sets

uses tools from these areas as well as from geometry, number theory,

and representation theory.

A group is often useful when it acts on a set or a structure. As we

shall explain, a group contains a difference set if and only if it acts in

a particular way on a nice structure called a symmetric design. Thus

finding a difference set is equivalent to finding an interesting group

action. Also, difference sets are of intrinsic interest because they yield

applications in communications and other areas.

So what is a difference set? If a finite group G is written ad-

ditively, a non-empty proper subset D of G is a (v, k, λ)-difference

set if |G| = v, |D| = k and there is an integer λ such that each

non-identity element of G can be expressed in exactly λ ways as a

difference d1 − d2 of elements of D. Equivalently, we require that

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http://dx.doi.org/10.1090/stml/067/01