Difference Sets: Connecting Algebra, Combinatorics, and Geometry (Student Mathematical Library) (Student Mathematical Library, 67)

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. Both authors are experienced teachers who have successfully supervised undergraduate research on difference sets 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. Both authors are experienced teachers who have successfully supervised undergraduate research on difference sets. Cover 1 Title page 4 Contents 8 Preface 12 Introduction 16 Designs 26 Automorphisms of designs 52 Introducing difference sets 60 Bruck-Ryser-Chowla theorem 86 Multipliers 102 Necessary group conditions 118 Difference sets from geometry 136 Families from Hadamard matrices 150 Representation theory 182 Group characters 212 Using algebraic number theory 248 Applications 268 Background 282 Notation 288 Hints and solutions to selected exercises 292 Bibliography 302 Index 308 Index of parameters 312 Back Cover 314 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. Emily H. Moore, Harriet S. Pollatsek. Includes Bibliographical References And Indexes.