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  • © 1957

Generators and Relations for Discrete Groups

Part of the book series: Ergebnisse der Mathematik und Ihrer Grenzgebiete. 1. Folge (MATHE1, volume 14)

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  • ISBN: 978-3-662-25739-5
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Table of contents (9 chapters)

  1. Front Matter

    Pages II-VIII
  2. Cyclic, Dicyclic and Metacyclic Groups

    • H. S. M. Coxeter, W. O. J. Moser
    Pages 1-12
  3. Systematic Enumeration of Cosets

    • H. S. M. Coxeter, W. O. J. Moser
    Pages 12-18
  4. Graphs, Maps and Cayley Diagrams

    • H. S. M. Coxeter, W. O. J. Moser
    Pages 18-32
  5. Abstract Crystallography

    • H. S. M. Coxeter, W. O. J. Moser
    Pages 33-52
  6. Hyperbolic Tessellations and Fundamental Groups

    • H. S. M. Coxeter, W. O. J. Moser
    Pages 52-61
  7. The Symmetric, Alternating, and other Special Groups

    • H. S. M. Coxeter, W. O. J. Moser
    Pages 61-82
  8. Modular and Linear Fractional Groups

    • H. S. M. Coxeter, W. O. J. Moser
    Pages 83-100
  9. Regular Maps

    • H. S. M. Coxeter, W. O. J. Moser
    Pages 100-117
  10. Groups Generated by Reflections

    • H. S. M. Coxeter, W. O. J. Moser
    Pages 117-133
  11. Back Matter

    Pages 134-155

About this book

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

  • University of Toronto, Canada

    H. S. M. Coxeter

  • University of Saskatchewan, Canada

    W. O. J. Moser

Bibliographic Information

  • Book Title: Generators and Relations for Discrete Groups

  • Authors: H. S. M. Coxeter, W. O. J. Moser

  • Series Title: Ergebnisse der Mathematik und Ihrer Grenzgebiete. 1. Folge

  • DOI: https://doi.org/10.1007/978-3-662-25739-5

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag Berlin Heidelberg 1957

  • Softcover ISBN: 978-3-662-23654-3

  • eBook ISBN: 978-3-662-25739-5

  • Edition Number: 1

  • Number of Pages: VIII, 155

  • Number of Illustrations: 2 b/w illustrations

  • Topics: Algebra

Buying options

eBook USD 79.99
Price excludes VAT (USA)
  • ISBN: 978-3-662-25739-5
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book USD 99.99
Price excludes VAT (USA)