A Course on Finite Groups

ISBN-13: 9781848828889 / Angielski / Miękka / 2010 / 324 str.

H. E. Rose

Opis

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 resume 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.

Recenzje

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)

Spis treści

Introduction—The Group Concept.- Elementary Group Properties.- Group Construction and Representation.- Homomorphisms.- Action and the Orbit–Stabiliser Theorem.- -Groups and Sylow Theory.- Products and Abelian Groups.- Groups of Order 24 Three Examples.- Series, Jordan–Hölder Theorem and the Extension Problem.- Nilpotency.- Solubility.- Simple Groups of Order Less than 10000.- Appendices A to E.

Blurb

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.