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