Finite Reflection Groups

  • L. C. Grove
  • C. T. Benson

Part of the Graduate Texts in Mathematics book series (GTM, volume 99)

Table of contents

  1. Front Matter
    Pages i-x
  2. L. C. Grove, C. T. Benson
    Pages 1-4
  3. L. C. Grove, C. T. Benson
    Pages 5-26
  4. L. C. Grove, C. T. Benson
    Pages 27-33
  5. L. C. Grove, C. T. Benson
    Pages 34-52
  6. L. C. Grove, C. T. Benson
    Pages 53-82
  7. L. C. Grove, C. T. Benson
    Pages 83-103
  8. L. C. Grove, C. T. Benson
    Pages 104-123
  9. L. C. Grove, C. T. Benson
    Pages 124-126
  10. Back Matter
    Pages 127-133

About this book

Introduction

Chapter 1 introduces some of the terminology and notation used later and indicates prerequisites. Chapter 2 gives a reasonably thorough account of all finite subgroups of the orthogonal groups in two and three dimensions. The presentation is somewhat less formal than in succeeding chapters. For instance, the existence of the icosahedron is accepted as an empirical fact, and no formal proof of existence is included. Throughout most of Chapter 2 we do not distinguish between groups that are "geo­ metrically indistinguishable," that is, conjugate in the orthogonal group. Very little of the material in Chapter 2 is actually required for the sub­ sequent chapters, but it serves two important purposes: It aids in the development of geometrical insight, and it serves as a source of illustrative examples. There is a discussion offundamental regions in Chapter 3. Chapter 4 provides a correspondence between fundamental reflections and funda­ mental regions via a discussion of root systems. The actual classification and construction of finite reflection groups takes place in Chapter 5. where we have in part followed the methods of E. Witt and B. L. van der Waerden. Generators and relations for finite reflection groups are discussed in Chapter 6. There are historical remarks and suggestions for further reading in a Post lude.

Keywords

Finite Groups Invariant Point group boundary element method classification construction development eXist finite group form formal proof group presentation proof

Authors and affiliations

  • L. C. Grove
    • 1
  • C. T. Benson
    • 1
  1. 1.Department of MathematicsUniversity of ArizonaTucsonUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4757-1869-0
  • Copyright Information Springer-Verlag New York 1985
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4419-3072-9
  • Online ISBN 978-1-4757-1869-0
  • Series Print ISSN 0072-5285
  • About this book