Blocks of Finite Groups and Their Invariants

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 clas...

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Main Author Sambale, Benjamin
Format eBook
LanguageEnglish
Published Cham Springer Nature 2014
Springer International Publishing AG
Springer International Publishing
Edition1
SeriesLecture Notes in Mathematics
Subjects
Algebra & number theory
Blocks (Group theory)
Finite groups
Group Theory and Generalizations
Invariants
Mathematics
Mathematics and Statistics
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Abstract 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 Most of the material is drawn from peer-reviewed journal articles, but there are also new previously unpublished results.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.
AbstractList 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 Most of the material is drawn from peer-reviewed journal articles, but there are also new previously unpublished results.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.
Author Sambale, Benjamin
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Snippet 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...
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SubjectTerms Algebra & number theory
Blocks (Group theory)
Finite groups
Group Theory and Generalizations
Invariants
Mathematics
Mathematics and Statistics
TableOfContents Index
Intro -- Introduction -- Contents -- List of Tables -- Part I Fundamentals -- Chapter 1 Definitions and Facts -- 1.1 Group Algebras and Blocks -- 1.2 Defect Groups and Characters -- 1.3 Brauer's Main Theorems -- 1.4 Covering and Domination -- 1.5 Fusion Systems -- 1.6 Subsections and Contributions -- 1.7 Centrally Controlled Blocks -- 1.8 Lower Defect Groups -- Chapter 2 Open Conjectures -- Part II General Results and Methods -- Chapter 3 Quadratic Forms -- Chapter 4 The Cartan Method -- 4.1 An Inequality -- 4.2 An Algorithm -- 4.3 The Inverse Cartan Method -- 4.4 More Inequalities -- Chapter 5 A Bound in Terms of Fusion Systems -- 5.1 The Case p=2 -- 5.2 The Case p&gt -- 2 -- Chapter 6 Essential Subgroups and Alperin's Fusion Theorem -- Chapter 7 Reduction to Quasisimple Groups and the Classification -- 7.1 Fong Reductions -- 7.2 Extensions of Nilpotent Blocks -- 7.3 Components -- 7.4 The Classification of the Finite Simple Groups -- 7.5 Blocks of p-Solvable Groups -- Part III Applications -- Chapter 8 Metacyclic Defect Groups -- 8.1 The Case p=2 -- 8.2 The Case p&gt -- 2 -- 8.2.1 Metacyclic, Minimal Non-abelian Defect Groups -- 8.2.2 One Family for p=3 -- Chapter 9 Products of Metacyclic Groups -- 9.1 D_2^n x C_2^m -- 9.2 D_2^n ∗ C_2^m -- 9.3 Q_2^n x C_2^m -- 9.4 SD_2^n x C_2^m -- Chapter 10 Bicyclic Groups -- 10.1 Fusion Systems -- 10.1.1 The Case P Non-cyclic -- 10.1.2 The Case P Cyclic -- 10.2 Blocks -- Chapter 11 Defect Groups of p-Rank 2 -- Chapter 12 Minimal Non-abelian Defect Groups -- 12.1 The Case p=2 -- 12.2 The Case p&gt -- 2 -- Chapter 13 Small Defect Groups -- 13.1 Results on the k(B)-Conjecture -- 13.2 2-Blocks of Defect 5 -- 13.3 Minimal Non-metacyclic Defect Groups -- Chapter 14 Abelian Defect Groups -- 14.1 The Brauer-Feit Bound -- 14.2 Abelian Groups of Small Rank -- Chapter 15 Blocks with Few Characters -- References
Title Blocks of Finite Groups and Their Invariants
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