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Blocks of finite groups and their invariants / Benjamin Sambale.

By: Material type: TextTextSeries: Lecture notes in mathematics ; 2127.Publication details: Switzerland : Springer, 2014.Description: xiii, 243 p. ; 24 cmISBN:
  • 9783319120058 (hard cover : alk. paper)
Subject(s): DDC classification:
  • 512.23 23 Sa187
Contents:
Part I Fundamentals -- 1 Definitions and Facts -- 2 Open Conjectures -- Part II General Results and Methods -- 3 Quadratic Forms -- 4 The Cartan Method -- 5 A Bound in Terms of Fusion Systems -- 6 Essential Subgroups and Alperin's Fusion Theorem -- 7 Reduction to Quasisimple Groups and the Classification -- Part III Applications -- 8 Metacyclic Defect Groups -- 9 Products of Metacyclic Groups -- 10 Bicyclic Groups -- 11 Defect Groups of p-Rank -- 12 Minimal Non-abelian Defect Groups -- 13 Small Defect Groups -- 14 Abelian Defect Groups -- 15 Blocks with Few Characters-- Rererences-- Index.
Summary: 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 Most of the material is drawn from peer-reviewed journal articles, but there are also new previously unpublished results.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.
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Includes bibliographical references and index.

Part I Fundamentals --
1 Definitions and Facts --
2 Open Conjectures --

Part II General Results and Methods --
3 Quadratic Forms --
4 The Cartan Method --
5 A Bound in Terms of Fusion Systems --
6 Essential Subgroups and Alperin's Fusion Theorem --
7 Reduction to Quasisimple Groups and the Classification --

Part III Applications --
8 Metacyclic Defect Groups --
9 Products of Metacyclic Groups --
10 Bicyclic Groups --
11 Defect Groups of p-Rank --
12 Minimal Non-abelian Defect Groups --
13 Small Defect Groups --
14 Abelian Defect Groups --
15 Blocks with Few Characters--
Rererences--
Index.

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.

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