© 2018

Orthogonal Latin Squares Based on Groups


  • Presents the first unified proof of the Hall–Paige conjecture

  • Discusses the actions of groups on designs derived from latin squares

  • Includes an extensive list of open problems on the construction and structure of orthomorphism graphs suitable for researchers and graduate students


Part of the Developments in Mathematics book series (DEVM, volume 57)

Table of contents

  1. Front Matter
    Pages i-xv
  2. Introduction

    1. Front Matter
      Pages 1-1
    2. Anthony B. Evans
      Pages 3-40
    3. Anthony B. Evans
      Pages 41-63
  3. Admissible Groups

    1. Front Matter
      Pages 65-65
    2. Anthony B. Evans
      Pages 91-114
    3. Anthony B. Evans
      Pages 169-199
  4. Orthomorphism Graphs of Groups

    1. Front Matter
      Pages 201-201
    2. Anthony B. Evans
      Pages 203-255
    3. Anthony B. Evans
      Pages 257-293
    4. Anthony B. Evans
      Pages 295-326
    5. Anthony B. Evans
      Pages 327-373
    6. Anthony B. Evans
      Pages 375-399
    7. Anthony B. Evans
      Pages 401-439
  5. Additional Topics

    1. Front Matter
      Pages 441-441
    2. Anthony B. Evans
      Pages 467-501

About this book


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

Authors and affiliations

  1. 1.Mathematics and StatisticsWright State UniversityDaytonUSA

About the authors

​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.

Bibliographic information

  • Book Title Orthogonal Latin Squares Based on Groups
  • Authors Anthony B. Evans
  • Series Title Developments in Mathematics
  • Series Abbreviated Title DEVM
  • DOI
  • Copyright Information Springer International Publishing AG, part of Springer Nature 2018
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-3-319-94429-6
  • Softcover ISBN 978-3-030-06850-9
  • eBook ISBN 978-3-319-94430-2
  • Series ISSN 1389-2177
  • Series E-ISSN 2197-795X
  • Edition Number 1
  • Number of Pages XV, 537
  • Number of Illustrations 90 b/w illustrations, 0 illustrations in colour
  • Topics Combinatorics
    Group Theory and Generalizations
  • Buy this book on publisher's site