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THE THEORY OF GROUPS THE THEORY OF GROUPS BY A.G.KUROSH TRANSLATED FROM THE RUSSIAN AND EDITED BY K. A. HIRSCH VOLUME ONE SECOND ENGLISH EDITION CHELSEA PUBLISHING COMPANY NEW YORK, N. Y. © CoPYRIGHT 1956, BY CHELSEA PuBLISHING CoMPANY © CoPYRIGHT 1960, BY CHELSEA PuBLISHING CoMPANY THE PRESENT WORK, PUBLISHED IN TWO VOLUMES, IS A TRANSLATION INTO ENGLISH, BY K. A. HIRSCH, OF THE SECOND RUSSIAN EDITION OF THE BOOK TEORIYA GRUPP BY A. G. KuRo~, WITH sUPPLEMENTARY MATERIAL BY THE TRANSLATOR LIBRARY oF CoNGREss CATALOG CARD NuMBER 60-8965 PRINTED IN THE UNITED STATES OF AMERICA TRANSLATOR'S PREFACE The book Teoriya Grupp by Professor A. G. Kuros has been widely acclaimed as the first modern text on the general theory of groups, with the major emphasis on infinite groups. An English translation of the work was, therefore, highly desirable. When I got in touch with the author and learned that the first Russian edition was out of print and that he was actively engaged in the preparations for a completely rewritten second edition, I decided to postpone my translation until the new book became available. This explains the delay between the first announcement and the actual issue of the present volume. (A German translation of the first Russian edition was published in 1953 by the Akademie-Verlag, Berlin.) In this translation I have followed the time-honoured maxim : "As literally as possible and as freely as necessary." Thus, while the book should read like an English text-book, it has, I hope, retained some of the flavour of the Russian original. A characteristic feature that the reader will notice is the author's sparing use of an elaborate symbolism and his reliance on a full verbal exposition of the mathematical argument. The changes I have made in the text can be described briefly as follows : ( i) Throughout the text I have distinguished between g G ( "g is an ele £ ment of G") and H c: G ("H is contained in G"). This distinction is not made in the Russian text, where the symbol c: is used in both meanings. Frequently I have changed the notation for certain subgroups, elements, sub scripts, indices, etc. to bring it into line with English usage. ( ii) I have slightly altered a few definitions (such as that of a free product and that of an element of infinite height) in order to avoid cumbersome case distinctions and to achieve more concise statements of some theorems. (iii) I have eliminated a number of misprints of the Russian text and have removed a few minor slips. Occasionally I have recast a proof where I thought it would lead to greater clarity. ( iv) The appendix notes, which are marked in the text by sans-serif su perior letters, contain a few additional remarks and some references to recent developments. This applies particularly to Parts Three and Four of the book, which are concerned with topics where progress is most rapid at present. ( v) I have tried to keep the bibliography up to date by adding to Volume II a separate list of references to relevant group-theoretical literature of the last few years. 5 6 TRANSLATOR'S PREFACE I may mention that the recent monograph by I. Kaplansky, Infinite Abelian Groups (University of Michigan Press, 1954), is an excellent sup plement to Part Two of this book, partly because the two books do not overlap too much in the material they cover, and partly because where they do overlap the two authors' different techniques make an interesting comparison. As the majority of the readers is likely to be in the United States, I have at the Publisher's request adopted American usage in spelling and termin ology. Thus, I talk of the center, of parentheses, and (somewhat reluctantly) of solvable groups, where in England it would have been the centre, brackets, and soluble groups. In the advanced parts of the book there is, quite understandably, much emphasis on the work of the very vigorous Russian group-theoretical school. The author is aware of this ; in a recent letter to me he writes : "The creation of the contemporary theory of groups was, and is, the work of a large world wide community of scholars, but the task of preparing a book that reflects the contemporary state of the theory of groups cannot be solved collectively." I hope that my translation will make English readers better acquainted with the trends of research in Russia and that in this way it will make a contribu tion to establishing a closer contact with our Russian colleagues. A final word about the use of the book as a text for graduate (or, in Eng land, advanced undergraduate) courses. I believe that in the hands of an experienced instructor the book will serve admirably as a text for students who have achieved a certain maturity of mathematical thinking. The instruc tor may have to make a few judicious omissions (of more difficult material) and additions (of further examples and exercises) . But in the theory of in finite groups good exercises of the right degree of difficulty are notoriously scarce-they tend to be either too trivial or too hard. During the last academic year, which I have spent as a Visiting Professor in the University of Colorado, at Boulder, I have covered both Part One and Part Two, each in a one semester three-hour course. I welcome this opportunity of expressing my thanks to the official agencies, the institutions, and the many colleagues who have helped to make my stay in the United States such a pleasant one. August, 1955 KURT A. HIRSCH PREFACE TO mE SECOND EDITION The author concluded his work on the first edition of this book in 1940, the proofs were read in the following year, and only the military circum stances of the time delayed the appearance of the book until 1944. Thus, nearly twelve years have passed since the book was completed. During these years the general theory of groups has undergone a remarkable change many problems have been solved, a number of new problems have arisen, and new directions of research have opened up, some of which now occupy a very conspicuous place in the theory of groups. In this rapid development of the theory of groups Soviet algebraists have played a prominent part. Young research workers have been systematically recruited, and continue to be recruited, into the Russian group-theoretical school, which was founded J. by 0. Schmidt. Their creative interests span almost all branches of the theory of groups, and in many directions the papers of Soviet scientists are among the leading ones. The first edition of the present book has also con tributed in some measure to the development of the group-theoretical studies -it might be mentioned that a typewritten copy was deposited in 1940 at the Institute for Mathematics and Mechanics of the University of Moscow and was accessible for study. When I began to prepare the second edition two years ago, I wanted to bring the book again up to the level our science had then attained. For this purpose I had to write virtually a new book. Not only does it differ from the old one in the planning of the material-many new sections have been added and many that were taken over from the old book have been completely revised-but hardly a single section has been transferred to the new book without some alterations. On the other hand, the increase in the volume of the book, which unfortunately could not be avoided, compelled me to omit a number of points that were in the old book and occasionally entire sections; however, they were of such a nature that their inclusion in the original book cannot be regarded as having been a mistake. I have therefore found it appropriate in some cases, when referring the reader to additional literature, to refer him also to the corresponding section of the first edition of the book. I must emphasize, however, that the new book has the old one as its basis and is very close to it in conception. This justifies me, I think, in keeping the old title for the book with the qualification ''Second Edition, Revised." I do not intend to give a complete survey of the book, but I shall point out the principal differences between its main parts and the corresponding 7 8 PREFACE TO SECOND EDITION parts of the first edition. Part One contains what one would naturally refer to as the elements of group theory. A thorough acquaintance with this material is assumed in all subsequent parts of the book. I mention one detail: The concept of the factor group and the homomorphism theorem appear in the book long before the concept of a normal subgroup is introduced. This interchange is not due to the needs of group theory itself and has been made only in order to expose the triviality of those all-too-numerous general izations of the group concept whose theory does not go much further than the homomorphism theorem. As is well known, this theorem can, in fact, be formulated and proved for sets with an arbitrary number of algebraic operations. The theory of abelian groups has been subjected to a drastic revision. This refers to primary abelian groups, in particular, whose theory has been considerably reorganized and enriched by the work of L. Y. Kulikov. As far as torsion-free abelian groups are concerned, the method of presenting the groups by systems of p-adic matrices has here been omitted, as it is of little help in the study of these groups ; instead, the theory of completely decomposable groups has been included. A considerable number of significant additions has been made in the theory of free groups and free products. In particular, some results recently obtained by B. H. Neumann and his collaborators have been incorporated in the book. In the theory of direct products of groups large re-dispositions have been undertaken; as a result of papers by the author and later by R. Baer, this theory is drawing appreciably closer to its completion. Therefore it was natural to deduce in the book the theorem of Schmidt (often also called theorem of Remak-Schmidt or Krull-Schmidt) from one of the much more general theorems obtained in recent years. This necessitated the develop ment of a large auxiliary apparatus and compelled me to combine the chapter on direct products with the chapter on lattices. In the first edition, only one section was devoted to group extensions. In the second edition it has grown into a whole chapter: this is due to the appearance of the cohomology theory in groups. Of course, even now the classification of extensions is far from having reached that degree of perfec tion which would allow the solving of any problem on extensions by a simple reference to this classification ; but the whole position cannot be compared to what it was twelve years ago. Particularly deep changes have occurred in the theory of soh,able and nilpotent infinite groups. The first edition of the book reflected only the first timid steps in this direction, and the relevant sections were included PREFACE TO SECOND EDITION 9 in the book more as a hint of subsequent developments than as an exposition of the results achieved at the time. To-day this is, in fact, one of the largest and richest branches of the theory of groups, a branch whose program can be expressed in these words : the study of groups which are closely related to abelian groups, under restrictions which in one sense or another are close to finiteness of the number of elements of the group. This new branch of the theory of groups has been created almost entirely by Soviet scientists. A special place belongs to S. N. Cernikov whose initi ative and creative contributions have determined the development of the researches in this domain to a remarkable degree. A number of results concerning very deep theorems have also been obtained by A. I. Mal'cev. Now a word about those parts of the theory of groups that have been omitted from the framework of the book. Among them there is above all the theory of finite groups. At the time when I worked on the first edition I set myself the task of showing that the theory of groups is not merely the theory of finite groups, and therefore the book contained almost nothing about finite groups in particular. This task can be regarded today as accom plished. Indeed, just the other way around: it has now become necessary to recall that the theory of finite groups is an important and integral part of the general theory of groups. Although some material on finite groups is now incorporated in this book, the above problem is by no means solved in it. It would be useful if one of the Soviet specialists on finite groups would write a small book devoted entirely to finite groups using the present book as a basis (that is, without expounding the elements of group theory over again). Even more urgent, perhaps, would be the writing of a book whose title could be given provisionally as the algebraic theory of groups of trans formations. It would have to contain the well-worked theory of permutation groups, the theory of groups of matrices, and also the general theory of representations of abstract groups. Isomorphic representations of groups by matrices, monomial groups and representations, the classi<,:al groups over an arbitrary field, and many other topics would also have to find a place in it. In a certain sense this is applied theory of groups. A systematic exposition of this entire branch of the theory of groups, using the results and metBods of the general theory of groups, would be very useful. The prerequisite knowledge that the reader of the book is assumed to possess has been indicated at the end of the Introduction to the First Edition. In addition, I might add that he should be acquainted with the concept of a ring and the simplest concepts connected with it.