An Introduction to Group Rings (Algebras and Applications, Volume 1) (Algebra and Applications)

group Rings Play A Central Role In The Theory Of Representations Of Groups And Are Very Interesting Algebraic Objects In Their Own Right. In Their Study, Many Branches Of Algebra Come To A Rich Interplay. This Book Takes The Reader From Beginning To Research Level And Contains Many Topics That, So Far, Were Only Found In Papers Published In Scientific Journals And, Whenever Possible, Offers New Proofs Of Known Results. It Also Includes Many Historical Notes And Some Applications. audience: This Book Will Be Of Interest To Mathematicians Working In The Area Of Group Rings And It Serves As An Introduction Of The Subject To Graduate Students. booknews sehgal (mathematical And Statistical Sciences, U. Of Alberta, Canada) And Milies (mathematics And Statistics, Universidade De São Paulo, Brasil) Intend This Text As An Introduction To The General Theory Of Group Rings, Which Will Take The Reader From Beginning To Research Level. The Text Requires A First-year Graduate Level Knowledge Of Algebra. Coverage Includes The Basics Of Group Representation Theory And Characters, The Connections Between This Theory And The Structure Of Group Algebras, The Isomorphism Problem, The Basic Properties Of Ideals In Group Rings, Algebraic Elements In Group Rings, And The Structure Of The Unit Group. For Mathematicians Working In The Area Of Group Rings, And For Use As An Introductory Text For Graduate Students. Annotation C. Book News, Inc., Portland, Or (booknews.com) to Group Rings by Cesar Polcino Milies Instituto de Matematica e Estatistica, Universidade de sao Paulo, sao Paulo, Brasil and Sudarshan K. Sehgal Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton. Canada SPRINGER-SCIENCE+BUSINESS MEDIA, B.V. A c.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-1-4020-0239-7 ISBN 978-94-010-0405-3 (eBook) DOI (http://10.1007/978-94-010-0405-3) 10.1007/978-94-010-0405-3 Printed an acid-free paper AII Rights Reserved 2002 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2002 Softcover reprint ofthe hardcover Ist edition 2002 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, inc1uding photocopying, recording Of by any information storage and retrieval system, without written permis sion from the copyright owner. Contents Preface ix 1 Groups 1 1.1 Basic Concepts . . . . . . . . . . . . 1 1.2 Homomorphisms and Factor Groups 10 1.3 Abelian Groups . 18 1.4 Group Actions, p-groups and Sylow Subgroups 21 1.5 Solvable and Nilpotent Groups 27 1.6 FC Groups . The famous memoir Reflections sur la resolution algebrique des equations, published by J.L. Lagrange in 1770, followed by papers of P. Ruffini and N.H. Abel, attracted the attention of working mathematicians to the concept of permutations (or linear substitutions, as they were called at that time).