Character Theory of Finite Groups

Author: I. Martin Isaacs

Publisher: Courier Corporation

ISBN: 9780486680149

Category: Mathematics

Page: 303

View: 8224

"The book is a pleasure to read. There is no question but that it will become, and deserves to be, a widely used textbook and reference." — Bulletin of the American Mathematical Society. Character theory provides a powerful tool for proving theorems about finite groups. In addition to dealing with techniques for applying characters to "pure" group theory, a large part of this book is devoted to the properties of the characters themselves and how these properties reflect and are reflected in the structure of the group. Chapter I consists of ring theoretic preliminaries. Chapters 2 to 6 and 8 contain the basic material of character theory, while Chapter 7 treats an important technique for the application of characters to group theory. Chapter 9 considers irreducible representations over arbitrary fields, leading to a focus on subfields of the complex numbers in Chapter 10. In Chapter 15 the author introduces Brauer’s theory of blocks and "modular characters." Remaining chapters deal with more specialized topics, such as the connections between the set of degrees of the irreducible characters and structure of a group. Following each chapter is a selection of carefully thought out problems, including exercises, examples, further results and extensions and variations of theorems in the text. Prerequisites for this book are some basic finite group theory: the Sylow theorems, elementary properties of permutation groups and solvable and nilpotent groups. Also useful would be some familiarity with rings and Galois theory. In short, the contents of a first-year graduate algebra course should be sufficient preparation.
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Representation Theory of Finite Groups

Author: Martin Burrow

Publisher: Courier Corporation

ISBN: 0486674878

Category: Mathematics

Page: 185

View: 4024

Concise, graduate-level exposition of the theory of finite groups, including the theory of modular representations. Topics include representation theory of rings with identity, representation theory of finite groups, applications of the theory of characters, construction of irreducible representations and modular representations. Rudiments of linear algebra and knowledge of group theory helpful prerequisites. Exercises. Bibliography. Appendix. 1965 edition.
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Representation Theory of Finite Groups

An Introductory Approach

Author: Benjamin Steinberg

Publisher: Springer Science & Business Media

ISBN: 9781461407768

Category: Mathematics

Page: 157

View: 9395

This book is intended to present group representation theory at a level accessible to mature undergraduate students and beginning graduate students. This is achieved by mainly keeping the required background to the level of undergraduate linear algebra, group theory and very basic ring theory. Module theory and Wedderburn theory, as well as tensor products, are deliberately avoided. Instead, we take an approach based on discrete Fourier Analysis. Applications to the spectral theory of graphs are given to help the student appreciate the usefulness of the subject. A number of exercises are included. This book is intended for a 3rd/4th undergraduate course or an introductory graduate course on group representation theory. However, it can also be used as a reference for workers in all areas of mathematics and statistics.
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Representation Theory of Finite Groups and Associative Algebras

Author: Charles W. Curtis,Irving Reiner

Publisher: American Mathematical Soc.

ISBN: 0821840665

Category: Mathematics

Page: 689

View: 6689

First published in 1962, this classic book remains a remarkably complete introduction to various aspects of the representation theory of finite groups. One of its main advantages is that the authors went far beyond the standard elementary representation theory, including a masterly treatment of topics such as general non-commutative algebras, Frobenius algebras, representations over non-algebraically closed fields and fields of non-zero characteristic, and integral representations. These and many other subjects are treated extremely thoroughly, starting with basic definitions and results and proceeding to many important and crucial developments. Numerous examples and exercises help the reader of this unsurpassed book to master this important area of mathematics.
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Finite Group Theory

Author: I. Martin Isaacs

Publisher: American Mathematical Soc.

ISBN: 0821843443

Category: Mathematics

Page: 350

View: 9900

The text begins with a review of group actions and Sylow theory. It includes semidirect products, the Schur-Zassenhaus theorem, the theory of commutators, coprime actions on groups, transfer theory, Frobenius groups, primitive and multiply transitive permutation groups, the simplicity of the PSL groups, the generalized Fitting subgroup and also Thompson's J-subgroup and his normal $p$-complement theorem. Topics that seldom (or never) appear in books are also covered. These include subnormality theory, a group-theoretic proof of Burnside's theorem about groups with order divisible by just two primes, the Wielandt automorphism tower theorem, Yoshida's transfer theorem, the ``principal ideal theorem'' of transfer theory and many smaller results that are not very well known. Proofs often contain original ideas, and they are given in complete detail. In many cases they are simpler than can be found elsewhere. The book is largely based on the author's lectures, and consequently, the style is friendly and somewhat informal. Finally, the book includes a large collection of problems at disparate levels of difficulty. These should enable students to practice group theory and not just read about it. Martin Isaacs is professor of mathematics at the University of Wisconsin, Madison. Over the years, he has received many teaching awards and is well known for his inspiring teaching and lecturing. He received the University of Wisconsin Distinguished Teaching Award in 1985, the Benjamin Smith Reynolds Teaching Award in 1989, and the Wisconsin Section MAA Teaching Award in 1993, to name only a few. He was also honored by being the selected MAA Polya Lecturer in 2003-2005.
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Representations and Characters of Groups

Author: Gordon James,Martin Liebeck

Publisher: Cambridge University Press

ISBN: 1139811053

Category: Mathematics

Page: N.A

View: 9046

This book provides a modern introduction to the representation theory of finite groups. Now in its second edition, the authors have revised the text and added much new material. The theory is developed in terms of modules, since this is appropriate for more advanced work, but considerable emphasis is placed upon constructing characters. Included here are the character tables of all groups of order less than 32, and all simple groups of order less than 1000. Applications covered include Burnside's paqb theorem, the use of character theory in studying subgroup structure and permutation groups, and how to use representation theory to investigate molecular vibration. Each chapter features a variety of exercises, with full solutions provided at the end of the book. This will be ideal as a course text in representation theory, and in view of the applications, will be of interest to chemists and physicists as well as mathematicians.
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A Course in the Theory of Groups

Author: Derek Robinson

Publisher: Springer Science & Business Media

ISBN: 1468401289

Category: Mathematics

Page: 481

View: 7864

" A group is defined by means of the laws of combinations of its symbols," according to a celebrated dictum of Cayley. And this is probably still as good a one-line explanation as any. The concept of a group is surely one of the central ideas of mathematics. Certainly there are a few branches of that science in which groups are not employed implicitly or explicitly. Nor is the use of groups confined to pure mathematics. Quantum theory, molecular and atomic structure, and crystallography are just a few of the areas of science in which the idea of a group as a measure of symmetry has played an important part. The theory of groups is the oldest branch of modern algebra. Its origins are to be found in the work of Joseph Louis Lagrange (1736-1813), Paulo Ruffini (1765-1822), and Evariste Galois (1811-1832) on the theory of algebraic equations. Their groups consisted of permutations of the variables or of the roots of polynomials, and indeed for much of the nineteenth century all groups were finite permutation groups. Nevertheless many of the fundamental ideas of group theory were introduced by these early workers and their successors, Augustin Louis Cauchy (1789-1857), Ludwig Sylow (1832-1918), Camille Jordan (1838-1922) among others. The concept of an abstract group is clearly recognizable in the work of Arthur Cayley (1821-1895) but it did not really win widespread acceptance until Walther von Dyck (1856-1934) introduced presentations of groups.
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Several Complex Variables with Connections to Algebraic Geometry and Lie Groups

Author: Joseph L. Taylor

Publisher: American Mathematical Soc.

ISBN: 082183178X

Category: Mathematics

Page: 507

View: 9222

This text presents an integrated development of the theory of several complex variables and complex algebraic geometry, leading to proofs of Serre's celebrated GAGA theorems relating the two subjects, and including applications to the representation theory of complex semisimple Lie groups. It includes a thorough treatment of the local theory using the tools of commutative algebra, an extensive development of sheaf theory and the theory of coherent analytic and algebraic sheaves, proofs of the main vanishing theorems for these categories of sheaves, and a complete proof of the finite dimensionality of the cohomology of coherent sheaves on compact varieties. The vanishing theorems have a wide variety of applications and these are covered in detail. Of particular interest are the last three chapters, which are devoted to applications of the preceding material to the study of the structure and representations of complex semisimple Lie groups.Included in this text are introductions to harmonic analysis, the Peter-Weyl theorem, Lie theory and the structure of Lie algebras, semisimple Lie algebras and their representations, algebraic groups and the structure of complex semisimple Lie groups. All of this culminates in Milicic's proof of the Borel-Weil-Bott theorem, which makes extensive use of the material developed earlier in the text. There are numerous examples and exercises in each chapter. This modern treatment of a classic point of view would be an excellent text for a graduate course on several complex variables, as well as a useful reference for the expert.
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Problems in Group Theory

Author: John D. Dixon

Publisher: Courier Corporation

ISBN: 0486459160

Category: Mathematics

Page: 176

View: 2842

265 challenging problems in all phases of group theory, gathered for the most part from papers published since 1950, although some classics are included.
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Representations of Compact Lie Groups

Author: T. Bröcker,T.tom Dieck

Publisher: Springer Science & Business Media

ISBN: 3662129183

Category: Mathematics

Page: 316

View: 9970

This introduction to the representation theory of compact Lie groups follows Herman Weyl’s original approach. It discusses all aspects of finite-dimensional Lie theory, consistently emphasizing the groups themselves. Thus, the presentation is more geometric and analytic than algebraic. It is a useful reference and a source of explicit computations. Each section contains a range of exercises, and 24 figures help illustrate geometric concepts.
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Group Theory and Its Application to Physical Problems

Author: Morton Hamermesh

Publisher: Courier Corporation

ISBN: 0486140393

Category: Science

Page: 544

View: 7496

One of the best-written, most skillful expositions of group theory and its physical applications, directed primarily to advanced undergraduate and graduate students in physics, especially quantum physics. With problems.
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Algebraic Function Fields and Codes

Author: Henning Stichtenoth

Publisher: Springer Science & Business Media

ISBN: 3540768785

Category: Mathematics

Page: 360

View: 5267

This book links two subjects: algebraic geometry and coding theory. It uses a novel approach based on the theory of algebraic function fields. Coverage includes the Riemann-Rock theorem, zeta functions and Hasse-Weil's theorem as well as Goppa' s algebraic-geometric codes and other traditional codes. It will be useful to researchers in algebraic geometry and coding theory and computer scientists and engineers in information transmission.
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Permutation Groups

Author: Donald S. Passman

Publisher: Courier Corporation

ISBN: 0486310914

Category: Mathematics

Page: 160

View: 4825

Lecture notes by a prominent authority provide a self-contained account of classification theorems. Includes work of Zassenhaus on Frobenius elements and sharply transitive groups, Huppert's theorem, more. 1968 edition.
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A First Course in Noncommutative Rings

Author: T.Y. Lam

Publisher: Springer Science & Business Media

ISBN: 1468404067

Category: Mathematics

Page: 397

View: 4438

One of my favorite graduate courses at Berkeley is Math 251, a one-semester course in ring theory offered to second-year level graduate students. I taught this course in the Fall of 1983, and more recently in the Spring of 1990, both times focusing on the theory of noncommutative rings. This book is an outgrowth of my lectures in these two courses, and is intended for use by instructors and graduate students in a similar one-semester course in basic ring theory. Ring theory is a subject of central importance in algebra. Historically, some of the major discoveries in ring theory have helped shape the course of development of modern abstract algebra. Today, ring theory is a fer tile meeting ground for group theory (group rings), representation theory (modules), functional analysis (operator algebras), Lie theory (enveloping algebras), algebraic geometry (finitely generated algebras, differential op erators, invariant theory), arithmetic (orders, Brauer groups), universal algebra (varieties of rings), and homological algebra (cohomology of rings, projective modules, Grothendieck and higher K-groups). In view of these basic connections between ring theory and other branches of mathemat ics, it is perhaps no exaggeration to say that a course in ring theory is an indispensable part of the education for any fledgling algebraist. The purpose of my lectures was to give a general introduction to the theory of rings, building on what the students have learned from a stan dard first-year graduate course in abstract algebra.
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Group Theory and Quantum Mechanics

Author: Michael Tinkham

Publisher: Courier Corporation

ISBN: 0486432475

Category: Science

Page: 340

View: 5941

Graduate-level text develops group theory relevant to physics and chemistry and illustrates their applications to quantum mechanics, with systematic treatment of quantum theory of atoms, molecules, solids. 1964 edition.
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Symmetry and the Monster

One of the Greatest Quests of Mathematics

Author: Mark Ronan

Publisher: Oxford University Press

ISBN: 0192807234

Category: History

Page: 255

View: 8878

Imagine a giant snowflake in 196,884 dimensions... This is the story of a mathematical quest that began two hundred years ago in revolutionary France, which led to the biggest collaboration ever between mathematicians across the world, and revealed the 'Monster' - a structure of beauty and complexity. And it is a story that is not yet over, for we have yet to understand the deep significance of the Monster - and its tantalising hints of connections with the physical structure of spacetime. Once we understand the full nature of the Monster, we may well have revealed a whole new and deeper understanding of the nature of our Universe.
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Discrete Harmonic Analysis

Representations, Number Theory, Expanders, and the Fourier Transform

Author: Tullio Ceccherini-Silberstein,Fabio Scarabotti,Filippo Tolli

Publisher: Cambridge University Press

ISBN: 1107182336

Category: Mathematics

Page: 666

View: 8543

A self-contained introduction to discrete harmonic analysis with an emphasis on the Discrete and Fast Fourier Transforms.
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Fourier Analysis on Finite Groups and Applications

Author: Audrey Terras

Publisher: Cambridge University Press

ISBN: 9780521457187

Category: Mathematics

Page: 442

View: 5195

A friendly introduction to Fourier analysis on finite groups, accessible to undergraduates/graduates in mathematics, engineering and the physical sciences.
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A Book of Abstract Algebra

Second Edition

Author: Charles C Pinter

Publisher: Courier Corporation

ISBN: 0486474178

Category: Mathematics

Page: 384

View: 9842

Accessible but rigorous, this outstanding text encompasses all of the topics covered by a typical course in elementary abstract algebra. Its easy-to-read treatment offers an intuitive approach, featuring informal discussions followed by thematically arranged exercises. This second edition features additional exercises to improve student familiarity with applications. 1990 edition.
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