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  • © 2012

Representation Theory of Finite Groups

An Introductory Approach

  • Concise focus enables the author to avoid prerequisites in analysis and topology

  • Provides a simple context for students to understand group representation theory

  • Author provides a gentle approach for students as well as the necessary motivation and exercises

  • Includes supplementary material: sn.pub/extras

Part of the book series: Universitext (UTX)

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eBook USD 59.99
Price excludes VAT (USA)
  • ISBN: 978-1-4614-0776-8
  • Instant PDF download
  • Readable on all devices
  • Own it forever
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  • Tax calculation will be finalised during checkout
Softcover Book USD 74.99
Price excludes VAT (USA)

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Table of contents (11 chapters)

  1. Front Matter

    Pages i-xiii
  2. Introduction

    • Benjamin Steinberg
    Pages 1-1
  3. Review of Linear Algebra

    • Benjamin Steinberg
    Pages 3-11
  4. Group Representations

    • Benjamin Steinberg
    Pages 13-25
  5. Character Theory and the Orthogonality Relations

    • Benjamin Steinberg
    Pages 27-50
  6. Fourier Analysis on Finite Groups

    • Benjamin Steinberg
    Pages 51-70
  7. Burnside’s Theorem

    • Benjamin Steinberg
    Pages 71-82
  8. Group Actions and Permutation Representations

    • Benjamin Steinberg
    Pages 83-96
  9. Induced Representations

    • Benjamin Steinberg
    Pages 97-109
  10. Another Theorem of Burnside

    • Benjamin Steinberg
    Pages 111-115
  11. Representation Theory of the Symmetric Group

    • Benjamin Steinberg
    Pages 117-129
  12. Probability and Random Walks on Groups

    • Benjamin Steinberg
    Pages 131-152
  13. Back Matter

    Pages 153-157

About this book

Representation Theory of Finite Groups presents group representation theory at a level accessible to advanced undergraduate students and beginning graduate students. The required background is maintained to the level of linear algebra, group theory, and very basic ring theory and avoids prerequisites in analysis and topology by dealing exclusively with finite groups. Module theory and Wedderburn theory, as well as tensor products, are deliberately omitted. Instead, an approach based on discrete Fourier Analysis is taken, thereby demanding less background from the reader.

The main topics covered in this text include character theory, the group algebra and Fourier analysis, Burnside's pq-theorem and the dimension theorem, permutation representations, induced representations and Mackey's theorem, and the representation theory of the symmetric group. For those students who have an elementary knowledge of probability and statistics, a chapter on random walks on finite groups serves as an illustration to link finite stochastics and representation theory. Applications to the spectral theory of graphs are given to help the student appreciate the usefulness of the subject and the author provides motivation and a gentle style throughout the text. A number of exercises add greater dimension to the understanding of the subject and some aspects of a combinatorial nature are clearly shown in diagrams.

This text will engage a broad readership due to the significance of representation theory in diverse branches of mathematics, engineering, and physics, to name a few. Its primary intended use is as a one semester textbook for a third or fourth year undergraduate course or an introductory graduate course on group representation theory. The content can also be of use as a reference to researchers in all areas of mathematics, statistics, and several mathematical sciences.

Keywords

  • Burnside's Theorem
  • character theory
  • group representation
  • induced representation
  • permutation representation

Reviews

From the reviews:

“The aim of the author is to give an introductory text on (ordinary) representation theory of finite groups which is accessible for advanced undergraduates already. And Steinberg manages this outstandingly well. … a book which is ideally suited for beginners in math as well as physicists, engineers and so on, who need a concise, well conceived and easy comprehensible introduction into representation theory.” (G. Kowol, Monatshefte für Mathematik, Vol. 167 (3-4), September, 2012)

“Steinberg … provides a one-semester course on representation theory with just linear algebra and a beginning course in abstract algebra (primarily group theory) as prerequisites. … the author covers most of the standard introductory topics in representation theory. The exercises provide more examples and further common results. It is the applications that Steinberg uses to motivate the subject that make this text both interesting and valuable. … Overall, a very user-friendly text with many examples and copious details. Summing Up: Recommended. Upper-division undergraduates through researchers/faculty.” (J. T. Zerger, Choice, Vol. 49 (11), August, 2012)

“The book consists of 157 pages spread over 11 chapters. … This book is an introductory course and it could be used by mathematicians and students who would like to learn quickly about the representation theory and character theory of finite groups, and for non-algebraists, statisticians and physicists who use representation theory.” (Jamshid Moori, Mathematical Reviews, Issue 2012 j)

“The required background as to this introductory course on group representations, is in the level of linear algebra, group theory and some ring theory. … the book under review is a welcome one for students at an advanced undergraduate or introductory graduate level course, also for those people like physicists, statisticians and non-algebraically oriented mathematicians who need representation theory in their work.” (R. W. van der Waall, Zentralblatt MATH, Vol. 1243, 2012)

“The author has, by combining clear writing with an accessible and minimal-prerequisite approach to group representations, created a book that may well help bring group representation theory into the undergraduate curriculum. This is an impressive and useful text, and should be looked at by anybody with an interest in the subject.” (Mark Hunacek, The Mathematical Association of America, February, 2012)

Authors and Affiliations

  • , Department of Mathematics, City College of New York, New York, USA

    Benjamin Steinberg

About the author

Benjamin Steinberg is a professor at City College of New York and the CUNY graduate center. He received my Ph.D. at the University of California, Berkeley. Steinberg is an algebraist interested in a broad range of areas including semigroups, geometric group theory and representation theory. Other research interests include automata theory, finite state Markov chains and algebras associated to etale groupoids. Steinberg is the co-author of a 2009 Springer publication in the SMM series entitled "The q-theory of Finite Semigroups". This book has had good pre- and post-publication reviews. Ben Steinberg is also an active editorial board member of the Semigroup Forum journal.

Bibliographic Information

  • Book Title: Representation Theory of Finite Groups

  • Book Subtitle: An Introductory Approach

  • Authors: Benjamin Steinberg

  • Series Title: Universitext

  • DOI: https://doi.org/10.1007/978-1-4614-0776-8

  • Publisher: Springer New York, NY

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer Science+Business Media, LLC 2012

  • Softcover ISBN: 978-1-4614-0775-1

  • eBook ISBN: 978-1-4614-0776-8

  • Series ISSN: 0172-5939

  • Series E-ISSN: 2191-6675

  • Edition Number: 1

  • Number of Pages: XIII, 157

  • Number of Illustrations: 4 b/w illustrations

  • Topics: Group Theory and Generalizations, Algebra

Buying options

eBook USD 59.99
Price excludes VAT (USA)
  • ISBN: 978-1-4614-0776-8
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book USD 74.99
Price excludes VAT (USA)