Systematic Enumeration of Cosets
E. H. Moore (1897) and many others have computed the index of a subgroup in an abstract group by systematically enumerating the cosets. Todd and Coxeter (1936) converted this into a mechanical technique, a useful tool with a wide range of applications. In
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Editorial Board: P. R. Halmos P. J. Hilton (Chairman) R. Remmert B. Szokefalvi-Nagy Advisors: L. V Ahlfors F L. Bauer A. Dold J. L. Doob S. Eilenberg K. W Gruenberg M. Kneser G. H. Muller M. M. Postnikov E. Sperner
H. S.M. Coxeter
W 0. J. Moser
Generators and Relations for Discrete Groups Fourth Edition With 54 Figures
Springer-Verlag Berlin Heidelberg GmbH 1980
H. S. M. Coxeter Department of Mathematics University of Toronto Toronto M5S 1A1, Ontario Canada W. 0. J. Moser Department of Mathematics McGill University Burnside Hall 805 Sherbrooke St. Montreal H3A 2K6, Quebec Canada
AMS Subject Classifications (1970): 05 B 45, 05 C 10, 05 C 20, 05 C 25, 55 A 05, 50 C 15, 20 E 40, 20 F 05, 20 F 10, 20 F 15, 20 H05, 20H 15, 20H20, 20 K05, 20G40
ISBN 978-3-662-21945-4 ISBN 978-3-662-21943-0 (eBook) DOI 10.1007/978-3-662-21943-0
Library of Congress Cataloging in Publication Data. Coxeter, Harold Scott Macdonald, 1907-. Generators and relations for discrete groups. (Ergebnisse der Mathematik und ihrer Grenzgebiete; Bd. 14). Bibliography: p. Includes index. 1. Discrete groups. 2. Groups, Theory of--Generators. 3. Groups, Theory of-Relations. I. Moser, W.O.J., joint author. II. Title. III. Series. QA17l.C7. 1979. 512'.2. 79-23198 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under §54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.
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Springer-Verlag Berlin Heidelberg 1957, 1964, 1972, 1980 Originally published by Springer-Verlag in 1980 Softcover reprint of the hardcover 4th edition 1980
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Preface to the First Edition When we began to consider the scope of this book, we envisaged a catalogue supplying at least one abstract definition for any finitelygenerated group that the reader might propose. But we soon realized that more or less arbitrary restrictions are necessary, because interesting groups are so numerous. For permutation groups of degree 8 or less (i.e.' .subgroups of es), the reader cannot do better than consult the tables of JosEPHINE BuRNS (1915), while keeping an eye open for misprints. Our own tables (on pages 134-142) deal with groups of low order, finite and infinite groups of congruent transformations, symmetric and alternating groups, linear fractional groups, and groups generated by reflections in real Euclidean space of any number of dimensions. The best substitute for a more extensive catalogue is the description (in Chapter 2) of a method whereby the reader can easily work out his own abstract definition for almost any given finite group. This method is sufficiently mechanical for the use of an electronic computer. There is also a topological method (Chapter 3), suitable not only for groups of low order
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