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Blocks of Finite Groups and Their Invariants

  • Book
  • © 2014

Overview

Authors:
  1. Benjamin Sambale

    1. Institut für Mathematik, Friedrich-Schiller-Universität Jena, Jena, Germany

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  • Covers a comprehensive range of the most recent literature on block theory
  • Contains new previously unpublished material
  • Can be used as a handy reference for blocks with given defect groups
  • Includes supplementary material: sn.pub/extras

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2127)

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Table of contents (15 chapters)

  1. Front Matter

    Pages i-xiii
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  2. Fundamentals

    1. Front Matter

      Pages 1-1
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    2. Definitions and Facts

      • Benjamin Sambale
      Pages 3-17

    3. Open Conjectures

      • Benjamin Sambale
      Pages 19-22

  3. General Results and Methods

    1. Front Matter

      Pages 23-23
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    2. Quadratic Forms

      • Benjamin Sambale
      Pages 25-32

    3. The Cartan Method

      • Benjamin Sambale
      Pages 33-46

    4. A Bound in Terms of Fusion Systems

      • Benjamin Sambale
      Pages 47-61

    5. Essential Subgroups and Alperin’s Fusion Theorem

      • Benjamin Sambale
      Pages 63-70

    6. Reduction to Quasisimple Groups and the Classification

      • Benjamin Sambale
      Pages 71-78

  4. Applications

    1. Front Matter

      Pages 79-79
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    2. Metacyclic Defect Groups

      • Benjamin Sambale
      Pages 81-94

    3. Products of Metacyclic Groups

      • Benjamin Sambale
      Pages 95-125

    4. Bicyclic Groups

      • Benjamin Sambale
      Pages 127-157

    5. Defect Groups of p-Rank 2

      • Benjamin Sambale
      Pages 159-165

    6. Minimal Non-abelian Defect Groups

      • Benjamin Sambale
      Pages 167-179

    7. Small Defect Groups

      • Benjamin Sambale
      Pages 181-203

    8. Abelian Defect Groups

      • Benjamin Sambale
      Pages 205-217

    9. Blocks with Few Characters

      • Benjamin Sambale
      Pages 219-227

  5. Back Matter

    Pages 229-246
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Keywords

About this book

Providing a nearly complete selection of up-to-date methods and results on block invariants with respect to their defect groups, this book covers the classical theory pioneered by Brauer, the modern theory of fusion systems introduced by Puig, the geometry of numbers developed by Minkowski, the classification of finite simple groups, and various computer assisted methods. In a powerful combination, these tools are applied to solve many special cases of famous open conjectures in the representation theory of finite groups. Most of the material is drawn from peer-reviewed journal articles, but there are also new previously unpublished results. In order to make the text self-contained, detailed proofs are given whenever possible. Several tables add to the text's usefulness as a reference. The book is aimed at experts in group theory or representation theory who may wish to make use of the presented ideas in their research.

Reviews

“It gives an excellent survey of the current state of knowledge with regards to the precise determination of the numerical invariants of blocks with a given defect group … . The book is a treasure trove of theorems and useful tables, and uses a wide variety of techniques, including the classification of finite simple groups and the GAP computer algebra package. It is the progress made without use of the classification, however, which is particularly impressive.” (Charles W. Eaton, Mathematical Reviews, February, 2016)

Authors and Affiliations

  • Institut für Mathematik, Friedrich-Schiller-Universität Jena, Jena, Germany

    Benjamin Sambale

Bibliographic Information

  • Book Title: Blocks of Finite Groups and Their Invariants

  • Authors: Benjamin Sambale

  • Series Title: Lecture Notes in Mathematics

  • DOI: https://doi.org/10.1007/978-3-319-12006-5

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer International Publishing Switzerland 2014

  • Softcover ISBN: 978-3-319-12005-8Published: 03 December 2014

  • eBook ISBN: 978-3-319-12006-5Published: 19 November 2014

  • Series ISSN: 0075-8434

  • Series E-ISSN: 1617-9692

  • Edition Number: 1

  • Number of Pages: XIII, 243

  • Number of Illustrations: 1 b/w illustrations

  • Topics: Group Theory and Generalizations

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