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Enumeration of Finite Groups


Part of Cambridge Tracts in Mathematics

  • Date Published: October 2007
  • availability: Available
  • format: Hardback
  • isbn: 9780521882170

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About the Authors
  • How many groups of order n are there? This is a natural question for anyone studying group theory, and this Tract provides an exhaustive and up-to-date account of research into this question spanning almost fifty years. The authors presuppose an undergraduate knowledge of group theory, up to and including Sylow's Theorems, a little knowledge of how a group may be presented by generators and relations, a very little representation theory from the perspective of module theory, and a very little cohomology theory - but most of the basics are expounded here and the book is more or less self-contained. Although it is principally devoted to a connected exposition of an agreeable theory, the book does also contain some material that has not hitherto been published. It is designed to be used as a graduate text but also as a handbook for established research workers in group theory.

    • The first book devoted to this exciting and vigorous area of modern group-theoretic research
    • Written by leading specialists in the field; contains hitherto unpublished material
    • Includes many open problems - ideal for graduate students in group theory
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    '… a welcome and well-written addition to the theory of finite groups.' EMS Newsletter

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    Product details

    • Date Published: October 2007
    • format: Hardback
    • isbn: 9780521882170
    • length: 294 pages
    • dimensions: 236 x 160 x 19 mm
    • weight: 0.532kg
    • contains: 37 exercises
    • availability: Available
  • Table of Contents

    1. Introduction
    Part I. Elementary Results:
    2. Some basic observations
    Part II. Groups of Prime Power Order:
    3. Preliminaries
    4. Enumerating p-groups: a lower bound
    5. Enumerating p-groups: upper bounds
    Part III. Pyber's Theorem:
    6. Some more preliminaries
    7. Group extensions and cohomology
    8. Some representation theory
    9. Primitive soluble linear groups
    10. The orders of groups
    11. Conjugacy classes of maximal soluble subgroups of symmetric groups
    12. Enumeration of finite groups with abelian Sylow subgroups
    13. Maximal soluble linear groups
    14. Conjugacy classes of maximal soluble subgroups of the general linear group
    15. Pyber's theorem: the soluble case
    16. Pyber's theorem: the general case
    Part IV. Other Topics:
    17. Enumeration within varieties of abelian groups
    18. Enumeration within small varieties of A-groups
    19. Enumeration within small varieties of p-groups
    20. Miscellanea
    21. Survey of other results
    22. Some open problems
    Appendix A. Maximising two equations.

  • Authors

    Simon R. Blackburn, Royal Holloway, University of London
    Simon Blackburn is a Professor of Pure Mathematics at Royal Holloway, University of London. He is also currently Head of Department in Mathematics at Royal Holloway. His mathematical interests include group theory, combinatorics and cryptography and some of the connections between these areas.

    Peter M. Neumann, The Queen's College, Oxford
    Peter Neumann is a Tutorial Fellow in Mathematics at Queen's College and University Lecturer in Pure Mathematics at Oxford University. His interests include many areas within algebra and group theory, including computational group theory, application of group theory in combinatorics, and nineteenth century history of group theory.

    Geetha Venkataraman, University of Delhi
    Gettha Venkataraman is a Senior Lecturer in Mathematics in St Stephen's College, at the University of Delhi. Her research interests involve enumerations of finite groups and other aspects of finite group theory. She is also interested in mathematics education for schools and the learning and teaching of mathematics at higher levels.

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