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THE THEORY OF GROUPS THE THEORY OF GROUPS HOMER BECHTELL University of New Hampshire ADDISON-WESLEY PUBLISHING COMPANY Reading, Massachusetts · Menlo Park, California · London · Don Mills, Ontario This book is in the ADDISON-WESLEY SERIES IN MATHEMATICS Consulting Editor: LYNN H. LOOMIS Copyright© 1971 by Addison-Wesley Publishing Company, Inc. Philippines copyright 1971 by Addison-Wesley Publishing Company, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, otherwise, without the prior written permission of the publisher. Printed in the United States America. Published simultaneously in Canada. Library of Congress Catalog Card No. 71-1361 To my parents PREFACE This book is designed to be used as a one-semester text. It evolved from a set oflectures that was to bridge the gap between the group theory presented in an introductory graduate algebra course and a serious pursuit of the subject. I feel that enough material is presented here for the student to make such a move comfortably. The style is moderate and an attempt has been made to develop as many topics as possible in a "natural" way. Hence, the book should be suitable for independent study. The overall approach and the methods of proof are varied but standard, and no novelties are introduced. Students have had little adjustment in the transition from this treatment to either special topics or reference works. Exercises are included with the original intent of the book in mind. For the most part, they are not difficult. In particular, this is true of the proofs of theorems and corollaries that have been left to the reader to complete. Most of the notation is standard. Since tensor products are not introduced in this text, the symbol used for the direct product should not cause confusion. I found that all of the topics and the association exercises can be covered in a one-semester program with the possible exception of the entirety of Chapter 8. The opening chapter summarizes the rudiments of group theory that are assumed known at this point. It is the material usually covered in an honors section of an undergraduate modern algebra course and a few topics that may have oeen delayed until the first year of graduate study. These possibly delayed topics are redeveloped here, but of course in more detail. A glance at the table of contents is enough to indicate the nature of the remaining portion. The bibliography consists of books rather than articles since this presentation wasn't intended to be a reference work. A brief introduction to category theory is found in the Appendix, but not as an afterthought. Undoubtedly many readers will find this useful as a re view. My experience has been that consistent use of the methods of category theory at this level created unanticipated difficulties in the students' under standing of the material. On the other hand, to apply the methods only to selected portions formed an inconsistent theme. So the inclusion of this introduction allows flexibility for an instructor who may prefer to begin with x Preface it and then to give alternative proofs at his discretion. In particular, one arrangement would be Appendix, Section 2.1, Chapter 9, and then Section 2.2 through Chapter 8. An introduction to the cohomology of groups is another omission. The reasons for this omission are much the same as those mentioned above for category theory. However, a treatment of this topic following a course based on the material in this text has been well received. I thank our students Stephen Bacon, Jiann Jer Chen, Paul Estes, Marshall Kotzen, Gail Lange, Paul Lepage, and Robert McDonald for their invalu able comments and suggestions. Acknowledgement is given also to Pro fessors David Burton and Richard Johnson for their helpful suggestions in the preparation of the manuscript. Durham, Nev.· Hampshire H.B. April 1971 CONTENTS Chapter 1 Basic Concepts and Notation . Chapter 2 Products, Direct Products, Direct Product with Amalgamated Subgroup, and Subdirect Products 2.1 Products and direct products . 8 2.2 Direct product with amalgamated subgroup 12 2.3 Subdirect products 16 Chapter 3 Splitting Extensions; Semidirect and Wreath Products 3.1 Products of subgroups 20 3.2 Extensions . 23 3.3 Splitting extensions 26 3.4 Wreath products . 29 Chapter 4 Theorems on Splitting; Hall Subgroups 4.1 On a theorem of Dixon . 35 4.2 Splitting theorems of Gaschlitz 39 4.3 On Hall n-subgroups 4.4 Additional comments 46 Chapter 5 Nilpotent Groups; the Frattini Subgroup 5.1 Nilpvtent groups . 49 5.2 The Sylow structure of a nilpotent group 53 5.3 The Frattini subgroup . 56 5.4 Additional remarks on the Frattini subgroup 62 Chapter 6 The Fitting Subgroup; SupersoiYable Groups 6.1 The Fitting subgroup 67 6.2 Supersolvable groups 69 xi