A Course on Finite Groups

H.E. Rose (auth.)1848828888, 9781848828889, 1848828896, 9781848828896

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: group constructions; homomorphisms and isomorphisms; actions; Sylow theory; products and Abelian groups; series; 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 problems, 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.

Table of contents :
Front Matter….Pages I-XII
Introduction—The Group Concept….Pages 1-10
Elementary Group Properties….Pages 11-40
Group Construction and Representation….Pages 41-65
Homomorphisms….Pages 67-90
Action and the Orbit–Stabiliser Theorem….Pages 91-111
p -Groups and Sylow Theory….Pages 113-137
Products and Abelian Groups….Pages 139-163
Groups of Order 24 Three Examples….Pages 165-185
Series, Jordan–Hölder Theorem and the Extension Problem….Pages 187-207
Nilpotency….Pages 209-227
Solubility….Pages 229-247
Simple Groups of Order Less than 10000….Pages 249-275
Appendices A to E….Pages 277-295
Back Matter….Pages 297-311