Graphs, Maps and Cayley Diagrams

The chief purpose of this chapter is to describe Cayley ’s representation of a group with given generators by a topological 1-complex or graph, whose vertices represent the elements of the group while certain sets of edges are associated with the generato

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HERAUSGEGEBEN VON

L.V.AHLFORS · R.BAER · R.COURANT· J.L.DOOB · S.EILENBERG P.R.HALMOS · T.NAKAYAMA · H.RADEMACHER F. K. SCHMIDT · B. SEGRE · E. SPERNER = = = = = = = N E U E FOLGE· HEFT 14 = = = = = = =

REIHE:

GRUPPENTHEORIE BESORGT VON

R.BAER

SPRINGER-VERLAG BERLIN HElDEiBERG GMBH 1957

GENERATORSAND RELATIONS FOR DISCRETE GROUPS H. S. M. COXETER AND

W. 0. J. MOSER

WITH 54 FIGURES

SPRINGER-YERLAG BERLIN HEIDELBERG GMBH 1957

ALLE RECHTE, INSBESONDERE DAS DER ÜBERSETZUNG IN FREMDE SPRACHEN, VORBEHALTEN OHNE AUSDRÜCKLICHE GENEHMIGUNG DES VERLAGES IST ES AUCH NICHT GESTATTET, DIESES BUCH ODER TEILE DARAUS AUF PHOTOMECHANISCHEM WEGE (PHOTOKOPIE, MIKROKOPIE) ZU VERVIELFÄLTIGEN © BY SPRINGER-VERLAG BERLIN HEIDELBERG 1957 URSPRÜNGliCH ERSCHIENEN BEl SPRINGER-VERLAG BERLIN HEIDELBERG GMBH 1957

ISBN 978-3-662-23654-3 ISBN 978-3-662-25739-5 (eBook) DOI 10.1007/978-3-662-25739-5

BRÜHLSCHE UNIVERSITÄTSDRUCKEREI GIESSEN

Preface 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 e 8), 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 134-143) deal with groups of low order, finiteandinfinite 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 foramoreextensive 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 but also for some infinite groups. This involves choosing a set of generators, constructing a certain graph (the Cayley diagram or DEHNsehe Gruppenbild), and embedding the graph into a surface. Cases in which the surface is a sphere or a plane are described in Chapter 4, where we obtain algebraically, and verify topologically, an abstract definition for each of the 17 space groups of two-dimensional crystallography. In Chapter 5, the fundamental groups of multiply-connected surfaces are exhibited as symmetry groups in the hyperbolic plane, the generators being translations or glide-reflections according as the surface is orientable or non-orientable. The next two chapters deal with special groups that have become famous for various reasons. In particular, certain generalizations of the polyhedral groups, scattered among the numerous papers of G. A. MILLER, are derived as members of a single family. The inclusion of a slightly different generalization in § 6. 7 is justified by its unexpec