Orthogonal Latin Squares Based on Groups
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 conj
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Anthony B. Evans
Orthogonal Latin Squares Based on Groups
Developments in Mathematics Volume 57
Series editors Krishnaswami Alladi, Gainesville, USA Pham Huu Tiep, Piscataway, USA Loring W. Tu, Medford, USA
More information about this series at http://www.springer.com/series/5834
Anthony B. Evans
Orthogonal Latin Squares Based on Groups
123
Anthony B. Evans Mathematics and Statistics Wright State University Dayton, OH, USA
ISSN 1389-2177 ISSN 2197-795X (electronic) Developments in Mathematics ISBN 978-3-319-94429-6 ISBN 978-3-319-94430-2 (eBook) https://doi.org/10.1007/978-3-319-94430-2 Library of Congress Control Number: 2018946725 Mathematics Subject Classification: 05-02, 05B15, 05B25, 05E18, 12E20, 20F99, 20N05, 11T06, 11T22 © Springer International Publishing AG, part of Springer Nature 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
To my wife Alma and to Leyla, Umair, Jasmina, and Ariya
Preface
Latin squares and mutually orthogonal sets of Latin squares (MOLS) have an old history predating Euler’s work in the late 1700s. With the emergence of the abstract concept of a group in the 1800s, Cayley observed that the multiplication/addition tables (Cayley tables) of groups are Latin squares. Latin squares and MOLS are used in several constructions of designs, notably nets, transversal designs, and affine and projective planes. In many constructions of designs from Latin squares and MOLS based on groups, the design is characterized by the action of the group on the design. The focus of this book is on orthomorphisms and complete mappings of finite groups. Orthomorphisms are permutations of the columns of the Cayley table of a group that yield L
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