Blocks of Finite Groups and Their Invariants

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

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      Pages 3-17

   3. Open Conjectures

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      Pages 19-22

3. General Results and Methods

   1. Front Matter

      Pages 23-23

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   2. Quadratic Forms

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      Pages 25-32

   3. The Cartan Method

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      Pages 33-46

   4. A Bound in Terms of Fusion Systems

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      Pages 47-61

   5. Essential Subgroups and Alperin’s Fusion Theorem

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      Pages 63-70

   6. Reduction to Quasisimple Groups and the Classification

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      Pages 71-78

4. Applications

   1. Front Matter

      Pages 79-79

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   2. Metacyclic Defect Groups

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      Pages 81-94

   3. Products of Metacyclic Groups

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      Pages 95-125

   4. Bicyclic Groups

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      Pages 127-157

   5. Defect Groups of p-Rank 2

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      Pages 159-165

   6. Minimal Non-abelian Defect Groups

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      Pages 167-179

   7. Small Defect Groups

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      Pages 181-203

   8. Abelian Defect Groups

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      Pages 205-217

   9. Blocks with Few Characters

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      Pages 219-227

5. Back Matter

   Pages 229-246

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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.

• 20C15,20C20,20C40
• Blocks of finite groups
• Defect groups
• Open conjectures
• Representation theory

“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)

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

  Benjamin Sambale

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