Orthomorphism Graphs of Groups
This book is about orthomorphisms and complete mappings of groups, and related constructions of orthogonal latin squares. It brings together, for the first time in book form, many of the results in this area. The aim of this book is to lay the foundations
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1535
Lecture Nates in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, ZUrich F. Takens, Groningen
1535
Anthony B. Evans
Orthomorphism Graphs of Groups
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Author Anthony B. Evans Department of Mathematics and Statistics Wright State University Dayton, OH 45435, USA
Mathematics Subject Classification (1991): Primary: 05-02, 05B 15 Secondary: 51E14, 51EI5
ISBN 3-540-56351-2 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-56351-2 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1992 Printed in Germany
Typesetting: Camera ready by author 46/3140-543210 Printed on acid-free paper
Preface The study of orthomorphism graphs of groups has its ongm in the use of orthomorphisms and complete mappings to construct mutually orthogonal sets of Latin squares. To help other mathematicians who wish to work in this area, a reference work is needed. None exists at present and work on the subject is scattered throughout the literature, often in a form that does not suggest any connection to orthomorphisms. In writing this monograph I have tried to do more than survey work done so far. In this monograph I have attempted to consolidate known results and applications, to create a unified body of knowledge and to provide other mathematicians with the tools needed to work in this area. I have tried to lay down the beginnings of a framework for the theory of orthomorphism graphs of groups and their applications, incorporating topics from algebra and geometry into this theory. As one of the hopes for this project was that it would stimulate research in this area, I have suggested many problems and directions for future research in this field, which should provide algebraists and geometers as well as other researchers in combinatorics with questions to work on. The material in this book should be accessible to any graduate student who has taken courses in group theory and field theory. I would like to thank Wright State University for its support during the writing of this manuscript, part of which was written while on sabbatical, and my colleagues Manley Perkel, who worked with me in generating orthomorphisms using Cayley, and Terry McKee, who read part of the manuscript, and the referees for their helpful suggestions.
Contents Preface.
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Contents
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Chapter 1: Introduction
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