# Kurosh: *The theory of groups* 2nd edition

The Russian text of

Here is a link to the Preface to the English translation of the Kurosh First Edition

We now quote from the Preface to the English translation of the

**Aleksandr G Kurosh**'s book*The theory of groups*was completed in 1940 but not published until 1944 because of problems caused by World War II. A second edition was brought out in 1952 and the second edition was translated into English and published by the Chelsea Publishing Company, New York, in 1953.Here is a link to the Preface to the English translation of the Kurosh First Edition

We now quote from the Preface to the English translation of the

**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 circumstances 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 bas 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 by 0 J 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 contributed 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 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 generalizations 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 Bernhard 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 development 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 perfection 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 solvable 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 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 initiative 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 Malcev.

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 accomplished. 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 transformations. 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 classical 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 methods 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. The bibliography has been revised, and supplemented by those papers published in recent years that have a bearing on the contents of the book.

Before and during the work on the second edition I received many comments and much advice-in letters, in personal talks, and in seminar meetings-from many Soviet algebraists. To all these fellow-mathematicians who have helped me with their advice I offer my sincere thanks.

Moscow, May 1952

A KUR0SH

Last Updated March 2006