Characters of Solvable Groups

Characters of Solvable Groups
Title Characters of Solvable Groups PDF eBook
Author I. Martin Isaacs
Publisher American Mathematical Soc.
Pages 384
Release 2018-05-23
Genre Mathematics
ISBN 1470434857

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This book, which can be considered as a sequel of the author's famous book Character Theory of Finite Groups, concerns the character theory of finite solvable groups and other groups that have an abundance of normal subgroups. It is subdivided into three parts: -theory, character correspondences, and M-groups. The -theory section contains an exposition of D. Gajendragadkar's -special characters, and it includes various extensions, generalizations, and applications of his work. The character correspondences section proves the McKay character counting conjecture and the Alperin weight conjecture for solvable groups, and it constructs a canonical McKay bijection for odd-order groups. In addition to a review of some basic material on M-groups, the third section contains an exposition of the use of symplectic modules for studying M-groups. In particular, an accessible presentation of E. C. Dade's deep results on monomial characters of odd prime-power degree is included. Very little of this material has previously appeared in book form, and much of it is based on the author's research. By reading a clean and accessible presentation written by the leading expert in the field, researchers and graduate students will be inspired to learn and work in this area that has fascinated the author for decades.

Representations of Solvable Groups

Representations of Solvable Groups
Title Representations of Solvable Groups PDF eBook
Author Olaf Manz
Publisher Cambridge University Press
Pages 318
Release 1993-09-16
Genre Mathematics
ISBN 0521397391

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Representation theory plays an important role in algebra, and in this book Manz and Wolf concentrate on that part of the theory which relates to solvable groups. The authors begin by studying modules over finite fields, which arise naturally as chief factors of solvable groups. The information obtained can then be applied to infinite modules, and in particular to character theory (ordinary and Brauer) of solvable groups. The authors include proofs of Brauer's height zero conjecture and the Alperin-McKay conjecture for solvable groups. Gluck's permutation lemma and Huppert's classification of solvable two-transive permutation groups, which are essentially results about finite modules of finite groups, play important roles in the applications and a new proof is given of the latter. Researchers into group theory, representation theory, or both, will find that this book has much to offer.

Characters and Blocks of Finite Groups

Characters and Blocks of Finite Groups
Title Characters and Blocks of Finite Groups PDF eBook
Author Gabriel Navarro
Publisher Cambridge University Press
Pages 301
Release 1998-05-07
Genre Mathematics
ISBN 0521595134

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This is a clear, accessible and up-to-date exposition of modular representation theory of finite groups from a character-theoretic viewpoint. After a short review of the necessary background material, the early chapters introduce Brauer characters and blocks and develop their basic properties. The next three chapters study and prove Brauer's first, second and third main theorems in turn. These results are then applied to prove a major application of finite groups, the Glauberman Z*-theorem. Later chapters examine Brauer characters in more detail. The relationship between blocks and normal subgroups is also explored and the modular characters and blocks in p-solvable groups are discussed. Finally, the character theory of groups with a Sylow p-subgroup of order p is studied. Each chapter concludes with a set of problems. The book is aimed at graduate students, with some previous knowledge of ordinary character theory, and researchers studying the representation theory of finite groups.

Characters and Blocks of Solvable Groups

Characters and Blocks of Solvable Groups
Title Characters and Blocks of Solvable Groups PDF eBook
Author James Cossey
Publisher Springer Nature
Pages 159
Release
Genre
ISBN 3031507061

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Character Theory of Finite Groups

Character Theory of Finite Groups
Title Character Theory of Finite Groups PDF eBook
Author I. Martin Isaacs
Publisher American Mathematical Soc.
Pages 322
Release 2006-11-21
Genre Mathematics
ISBN 0821842293

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Character theory is a powerful tool for understanding finite groups. In particular, the theory has been a key ingredient in the classification of finite simple groups. Characters are also of interest in their own right, and their properties are closely related to properties of the structure of the underlying group. The book begins by developing the module theory of complex group algebras. After the module-theoretic foundations are laid in the first chapter, the focus is primarily on characters. This enhances the accessibility of the material for students, which was a major consideration in the writing. Also with students in mind, a large number of problems are included, many of them quite challenging. In addition to the development of the basic theory (using a cleaner notation than previously), a number of more specialized topics are covered with accessible presentations. These include projective representations, the basics of the Schur index, irreducible character degrees and group structure, complex linear groups, exceptional characters, and a fairly extensive introduction to blocks and Brauer characters. This is a corrected reprint of the original 1976 version, later reprinted by Dover. Since 1976 it has become the standard reference for character theory, appearing in the bibliography of almost every research paper in the subject. It is largely self-contained, requiring of the reader only the most basic facts of linear algebra, group theory, Galois theory and ring and module theory.

A Course in Finite Group Representation Theory

A Course in Finite Group Representation Theory
Title A Course in Finite Group Representation Theory PDF eBook
Author Peter Webb
Publisher Cambridge University Press
Pages 339
Release 2016-08-19
Genre Mathematics
ISBN 1107162394

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This graduate-level text provides a thorough grounding in the representation theory of finite groups over fields and rings. The book provides a balanced and comprehensive account of the subject, detailing the methods needed to analyze representations that arise in many areas of mathematics. Key topics include the construction and use of character tables, the role of induction and restriction, projective and simple modules for group algebras, indecomposable representations, Brauer characters, and block theory. This classroom-tested text provides motivation through a large number of worked examples, with exercises at the end of each chapter that test the reader's knowledge, provide further examples and practice, and include results not proven in the text. Prerequisites include a graduate course in abstract algebra, and familiarity with the properties of groups, rings, field extensions, and linear algebra.

Characters of Solvable Groups

Characters of Solvable Groups
Title Characters of Solvable Groups PDF eBook
Author I. Martin Isaacs
Publisher
Pages
Release 2018
Genre MATHEMATICS
ISBN 9781470447250

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This book, which can be considered as a sequel of the author's famous book Character Theory of Finite Groups, concerns the character theory of finite solvable groups and other groups that have an abundance of normal subgroups. It is subdivided into three parts: \pi-theory, character correspondences, and M-groups. The \pi-theory section contains an exposition of D. Gajendragadkar's \pi-special characters, and it includes various extensions, generalizations, and applications of his work. The character correspondences section proves the McKay character counting conjecture and the Alperin weight c.