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Generators and Relations for Discrete Groups

  • H. S. M. Coxeter
  • W. O. J. Moser

Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE1, volume 14)

Table of contents

  1. Front Matter
    Pages II-VIII
  2. H. S. M. Coxeter, W. O. J. Moser
    Pages 1-12
  3. H. S. M. Coxeter, W. O. J. Moser
    Pages 12-18
  4. H. S. M. Coxeter, W. O. J. Moser
    Pages 18-32
  5. H. S. M. Coxeter, W. O. J. Moser
    Pages 33-52
  6. H. S. M. Coxeter, W. O. J. Moser
    Pages 52-61
  7. H. S. M. Coxeter, W. O. J. Moser
    Pages 61-82
  8. H. S. M. Coxeter, W. O. J. Moser
    Pages 83-100
  9. H. S. M. Coxeter, W. O. J. Moser
    Pages 100-117
  10. H. S. M. Coxeter, W. O. J. Moser
    Pages 117-133
  11. Back Matter
    Pages 134-155

About this book

Introduction

When we began to consider the scope of this book, we envisaged a catalogue supplying at least one abstract definition for any finitely­ generated 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 ), the reader cannot do better than consult the 8 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.

Keywords

Permutation algebra finite group transformation

Authors and affiliations

  • H. S. M. Coxeter
    • 1
  • W. O. J. Moser
    • 2
  1. 1.University of TorontoCanada
  2. 2.University of SaskatchewanCanada

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-662-25739-5
  • Copyright Information Springer-Verlag Berlin Heidelberg 1957
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-662-23654-3
  • Online ISBN 978-3-662-25739-5
  • Buy this book on publisher's site
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