Difference sets belong both to group theory and to combinatorics. Studying them requires tools from geometry, number theory, and representation theory. This book lays a foundation for these topics, including a primer on representations and characters of finite groups. It makes the research literature on difference sets accessible to students who have studied linear algebra and abstract algebra, and it prepares them to do their own research.

This text is suitable for an undergraduate capstone course, since it illuminates the many links among topics that the students have already studied. To this end, almost every chapter ends with a coda highlighting the main ideas and emphasizing mathematical connections. This book can also be used for self-study by anyone interested in these connections and concrete examples.

An abundance of exercises, varying from straightforward to challenging, invites the reader to solve puzzles, construct proofs, and investigate problems—by hand or on a computer. Hints and solutions are provided for selected exercises, and there is an extensive bibliography. The last chapter introduces a number of applications to real-world problems and offers suggestions for further reading.

Emily H. Moore, Grinnell College, Grinnell, IA

Harriet S. Pollatsek, Mount Holyoke College, South Hadley, MA

Chapter 1. Introduction
Chapter 2. Designs
Chapter 3. Automorphisms of designs
Chapter 4. Introducing difference sets
Chapter 5. Bruck-Ryser-Chowla theorem
Chapter 6. Multipliers
Chapter 7. Necessary group conditions
Chapter 8. Difference sets from geometry
Chapter 9. Families from Hadamard matrices
Chapter 10. Representation theory
Chapter 11. Group characters
Chapter 12. Using algebraic number theory
Chapter 13. Applications
Appendix A. Background
Appendix B. Notation
Appendix C. Hints and solutions to selected exercises