Universitext

By: H.E. Rose

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Paperback | 4 February 2010

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324 Pages

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22.86 x 14.61 x 1.91

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A Course on Finite Groups introduces the fundamentals of group theory to advanced undergraduate and beginning graduate students. Based on a series of lecture courses developed by the author over many years, the book starts with the basic definitions and examples and develops the theory to the point where a number of classic theorems can be proved. The topics covered include: Lagrange's theorem; group constructions; homomorphisms and isomorphisms; actions; Sylow theory, products and Abelian groups; series, and nilpotent and soluble groups; and an introduction to the classification of the finite simple groups.

A number of groups are described in detail and the reader is encouraged to work with one of the many computer algebra packages available to construct and experience "actual" groups for themselves in order to develop a deeper understanding of the theory and the significance of the theorems. Numerous exercises, of varying levels of difficulty, help to test understanding.

Industry Reviews

From the reviews:

"This is a self-contained introduction to the theory of finite groups. The treatment is exhaustive, from the elementary basic results up to characters and representations of finite groups ... . main results are accompanied by several well-chosen examples, and there are many computations with groups of small order, giving the reader a sense of concreteness. ... well-written book, not too wordy nor too terse. Concepts and results are illustrated with examples, and the problem sets at the end of every chapter nicely complement the theory." (Felipe Zaldivar, The Mathematical Association of America, March, 2010)

"This book is an introduction to the finite group theory for undergraduate and beginning graduate mathematicians." (Anatoli Kondrat'ev, Zentralblatt MATH, Vol. 1200, 2011)

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You Can Find This Book In

Non-Fiction Mathematics Groups & Group Theory Algebra Science Science in General

Introduction-The Group Concept	p. 1
Elementary Group Properties	p. 11
Basic Definitions	p. 11
Examples	p. 17
Subgroups, Cosets and Lagrange's Theorem	p. 24
Normal Subgroups	p. 30
Problems	p. 34
Group Construction and Representation	p. 41
Permutations	p. 42
Permutation Groups	p. 48
Matrix Groups	p. 52
Group Presentation	p. 55
Problems	p. 59
Homomorphisms	p. 67
Homomorphisms and Isomorphisms	p. 69
Isomorphism Theorems	p. 74
Cyclic Groups	p. 79
Automorphism Groups	p. 81
Problems	p. 84
Action and the Orbit-Stabiliser Theorem	p. 91
Actions	p. 92
Three Important Examples	p. 99
Problems	p. 107
p-Groups and Sylow Theory	p. 113
Finite p-Groups	p. 144
Sylow Theory	p. 119
Applications	p. 126
Problems	p. 131
Products and Abelian Groups	p. 139
Direct Products	p. 140
Finite Abelian Groups	p. 146
Semi-direct Products	p. 151
Problems	p. 159
Groups of Order 24, Three Examples	p. 165
Symmetric Group S₄	p. 165
Special Linear Group SL₂(3)	p. 172
Exceptional Group E	p. 177
Problems	p. 183
Series, Jordan-Hölder Theorem and the Extension Problem	p. 187
Composition Series and the Jordan-Hölder Theorem	p. 188
Extension Problem	p. 196
Problems	p. 205
Nilpotency	p. 209
Nilpotent Groups	p. 210
Frattini and Fitting Subgroups	p. 217
Problems	p. 223
Solubility	p. 229
Soluble Groups	p. 230
Hall's Theorems and Solubility Conditions	p. 236
Problems	p. 243
Simple Groups of Order Less than 10000	p. 249
Steiner Systems	p. 250
Linear Groups	p. 254
Unitary Groups	p. 265
Mathieu Groups	p. 267
Problems	p. 270
Appendices A to E	p. 277
Set Theory	p. 277
Number Theory	p. 284
Data on Groups of Order 24	p. 289
Numbers of Groups with Order up to 520	p. 293
Representations of L₂(q) with Order < 10000	p. 295
Bibliography	p. 297
Notation Index	p. 301
Symbol Index	p. 301
Notation for Classes of Groups	p. 303
Notation for Individual Groups	p. 304
Index	p. 305
Table of Contents provided by Ingram. All Rights Reserved.

Universitext

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