A Course on Finite Groups

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Springer Science & Business Media, Dec 16, 2009 - Mathematics - 311 pages
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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.

A brief resumé of the basic set theory and number theory required for the text is provided in an appendix, and a wealth of extra resources is available online at www.springer.com, including: hints and/or full solutions to all of the exercises; extension material for many of the chapters, covering more challenging topics and results for further study; and two additional chapters providing an introduction to group representation theory.

 

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Contents

IntroductionThe Group Concept
1
Elementary Group Properties
11
Group Construction and Representation
41
Homomorphisms
67
Action and the OrbitStabiliser Theorem
91
pGroups and Sylow Theory
112
Products and Abelian Groups
139
Groups of Order 24 Three Examples
164
Nilpotency
208
Solubility
229
Simple Groups of Order Less than 10000
248
Appendices A to E
277
Bibliography
296
Notation Index
301
Index
305
Copyright

Series JordanHölder Theorem and the Extension Problem
187

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