GROUP REPRESENTATIONS Volume 1 Part A: Background Material NORTH-HOLLAND MATHEMATICS STUDIES 175 {Continuation of the Notas de Matematica) Editor: Leopoldo NACHBIN Centro Brasileiro de Pesquisas Fisicas Rio de Janeiro, Brazil and University of Rochester New York, U.S.A. NORTH-HOLLAND - AMSTERDAM • LONDON • NEW YORK• TOKYO GROUP REPRESENTATIONS Volume 1 Part A: Background Material Gregory KARPILOVSKY Department of Mathematics California State University Chico, CA, U.S.A. ~ ~ ffl 1992 NORTH-HOLLAND - AMSTERDAM • LONDON • NEW YORK• TOKYO ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands Library of Congress Cataloging-in-Publication Data Karpilovsky, Gregory, 1940 - Group representations/ Gregory Karpilovsky. p. cm ... (North-Holland mathematics studies;l 75) Includes bibliographical references and index. Contents; v., pt .. A. Background material, pt. B. Introduction to group representations and characters. ISBN 0-444 -88632-X (set) 1. Representations of groups. I. Title. II. Series. QAl 76.K37 1992 92-14786 512' .2-dc20 CIP ISBN: 0 444 88632 X (Set: Part A and B) © 1992 ELSEVIER SCIENCE PUBLISHERS B.V. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science Publishers B.V., Copy right & Permissions Department, P.O. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the publisher. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. This book is printed on acid-free paper. Printed in The Netherlands To my wife Helen, who typed this lengthy manuscript and remained cheerful throughout vii Preface The present volume is the first of a multi-volume treatise on group repre sentations. The principal object of these volumes is to provide, in a self contained manner, a comprehensive coverage of the mainstream of group representation theory. The appropriate audience for these volumes consists of aspiring graduate students and mature mathematicians working in the field of group representations. No mathematical knowledge is presupposed beyond the rudiments of abstract algebra, set theory and field theory; how ever, a certain maturity in mathematical reasoning is required. Apart from a few obvious exceptions ( e.g. odd order theorem, classification of finite sim ple groups, etc.), the volumes are entirely self-contained. The style of the presentation is informal : the author is not afraid to repeat definitions and formulas when they are needed. Many sections begin with a nontechnical description of what is about to be done. A special effort has been made to render the exposition transparent. Due to bookbinding considerations, Volume 1 is split into two parts, Part A and Part B. However, mathematically, the material is divided into three parts. The first part provides background material for all volumes. The second part is devoted to introduction to group representations, and the third part to introduction to group characters. Many of the results are pre sented in their greatest generality and much of the material discussed is not conveniently available in other monographs on the subject. A systematic description of the material is supplied by the introductions to individual chapters and therefore will not be repeated here. A word about notation. As is customary, Theorem 2.3.4 denotes the fourth result in Section 3 of Chapter 2; however, for simplicity, all references to this result within Chapter 2 itself, are designated as Theorem 3.4. I am indebted to my wife Helen, who not only typed the entire volume, but encouraged me (always with patience and good humour) throughout the entire project. California State University, Chico G. Karpilovsky November, 1991 Contents Preface Vll Part I Background Material 1 1. Rings and Modules 3 1.1. Notation and terminology 3 1.2. Preliminary results 8 1.3. Artinian and noetherian modules and rings 13 1.4. Semisimple modules 22 1.5. The radical and socle of modules and rings 26 1.5.A. The radical and socle of modules 26 1.5.B. The Jacobson radical 31 1.5.C. The Jacobson radical and idempotents 39 1.6. Idempotent lifting theory 42 1.6.A. General results 42 1.6 .B. Semiregular rings 46 1.6.C. Algebras over complete rings 52 1.7. Azumaya's theorems 55 1.8. Local rings 60 1.9. Endomorphism algebras 61 1.10. Strongly indecomposable modules 65 1.10.A. Basic properties and characterizations 65 1.10.B. Azumaya's decomposition theorem 70 1.10.C. The Krull-Schmidt theorem 71 1.11. Direct decompositions and blocks 73 1.12. Matrix rings 80 2. Artinian and Semilocal Rings 89 2.1. Semiprimitive artinian rings 89 2.2. Semilocal rings 92 2.3. Artinian rings 100 2.3.A. A characterization 100