Orthogonal Latin Squares Based on Groups

1. Front Matter

   Pages i-xv

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2. Introduction

   1. Front Matter

      Pages 1-1

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   2. Latin Squares Based on Groups

      • Anthony B. Evans

      Pages 3-40

   3. When Is a Latin Square Based on a Group?

      • Anthony B. Evans

      Pages 41-63

3. Admissible Groups

   1. Front Matter

      Pages 65-65

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   2. The Existence Problem for Complete Mappings: The Hall-Paige Conjecture

      • Anthony B. Evans

      Pages 67-90

   3. Some Classes of Admissible Groups

      • Anthony B. Evans

      Pages 91-114

   4. The Groups GL(n, q), SL(n, q), PGL(n, q), and PSL(n, q)

      • Anthony B. Evans

      Pages 115-145

   5. Minimal Counterexamples to the Hall-Paige Conjecture

      • Anthony B. Evans

      Pages 147-167

   6. A Proof of the Hall-Paige Conjecture

      • Anthony B. Evans

      Pages 169-199

4. Orthomorphism Graphs of Groups

   1. Front Matter

      Pages 201-201

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   2. Orthomorphism Graphs of Groups

      • Anthony B. Evans

      Pages 203-255

   3. Elementary Abelian Groups. I

      • Anthony B. Evans

      Pages 257-293

   4. Elementary Abelian Groups. II

      • Anthony B. Evans

      Pages 295-326

   5. Extensions of Orthomorphism Graphs

      • Anthony B. Evans

      Pages 327-373

   6. ω(G) for Some Classes of Nonabelian Groups

      • Anthony B. Evans

      Pages 375-399

   7. Groups of Small Order

      • Anthony B. Evans

      Pages 401-439

5. Additional Topics

   1. Front Matter

      Pages 441-441

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   2. Projective Planes from Complete Sets of Orthomorphisms

      • Anthony B. Evans

      Pages 443-465

   3. Related Topics

      • Anthony B. Evans

      Pages 467-501

This monograph presents a unified exposition of latin squares and mutually orthogonal sets of latin squares based on groups. Its focus is on orthomorphisms and complete mappings of finite groups, while also offering a complete proof of the Hall–Paige conjecture. The use of latin squares in constructions of nets, affine planes, projective planes, and transversal designs also motivates this inquiry.  


The text begins by introducing fundamental concepts, like the tests for determining whether a latin square is based on a group, as well as orthomorphisms and complete mappings. From there, it describes the existence problem for complete mappings of groups, building up to the proof of the Hall–Paige conjecture. The third part presents a comprehensive study of orthomorphism graphs of groups, while the last part provides a discussion of Cartesian projective planes, related combinatorial structures, and a list of open problems.  


Expanding the author’s 1992 monograph, Orthomorphism Graphs of Groups, this book is an essential reference tool for mathematics researchers or graduate students tackling latin square problems in combinatorics. Its presentation draws on a basic understanding of finite group theory, finite field theory, linear algebra, and elementary number theory—more advanced theories are introduced in the text as needed. 

• Orthomorphism
• Complete mapping
• Latin square
• MOLS
• Difference matrix
• Orthogonality
• Finite group
• Finite field
• combinatorics

• Mathematics and Statistics, Wright State University, Dayton, USA

  Anthony B. Evans

​Anthony B. Evans is Professor of Mathematics at Wright State University in Dayton, Ohio. Since the mid 1980s, his primary research has been on orthomorphisms and complete mappings of finite groups and their applications. These mappings arise in the study of mutually orthogonal latin squares that are derived from the multiplication tables of finite groups. As an offshoot of this research, he has also worked on graph representations. His previous book, Orthomorphism Graphs of Groups (1992), appeared in the series, Lecture Notes in Mathematics.

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