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 emphasising 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. Both authors are experienced teachers who have successfully supervised undergraduate research on difference sets.
Emily H. Moore, Grinnell College, IA, USA. Harriet S. Pollatsek, Mount Holyoke College, South Hadley, MA, USA.
Table of Contents:Introduction Designs Automorphisms of designs Introducing difference sets Bruck-Ryser-Chowla theorem Multipliers Necessary group conditions Difference sets from geometry Families from Hadamard matrices Representation theory Group characters Using algebraic number theory Applications Background Notation Hints and solutions to selected exercises Bibliography Index Index of parameters