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Orthogonal Latin Squares Based on Groups

  • Anthony B. Evans

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
    3. Anthony B. Evans
      Pages 503-520
  6. Back Matter
    Pages 521-537

About this book

Introduction

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. 

Keywords

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

Authors and affiliations

  • Anthony B. Evans
    • 1
  1. 1.Mathematics and StatisticsWright State UniversityDaytonUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-94430-2
  • Copyright Information Springer International Publishing AG, part of Springer Nature 2018
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-94429-6
  • Online ISBN 978-3-319-94430-2
  • Series Print ISSN 1389-2177
  • Series Online ISSN 2197-795X
  • Buy this book on publisher's site