We are drawn to the study of difference sets because this topic “be-
longs both to group theory and to combinatorics and . . . uses tools
from these areas as well as from geometry, number theory, and rep-
resentation theory” (quoting from the opening of Chapter 1). Each
of us has supervised undergraduate research on difference sets. Our
original goal in writing this book was to collect in one place the ma-
terial beyond a one-semester abstract algebra course required to pre-
pare our students for these research projects. However, the links to
many parts of mathematics led to our current, broader aim: not only
to serve prospective undergraduate researchers but also to provide
a rich text for a senior seminar or capstone course in mathematics.
With this expanded goal in mind, we highlight these mathematical
We never intended our book to be a comprehensive survey of
difference sets. However, we hope it will encourage students to explore
the literature on difference sets and give them a solid foundation so
they can do so successfully.
We assume student readers have taken an abstract algebra course.1
We show them concrete examples of some algebraic ideas they studied
there, and we apply and extend these concrete instances in a variety
A includes the background we need from prior courses, and specific
results are cited using the notation A.x.