Kurosh: The theory of groups 1st edition

The Russian text of 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 Second Edition

We now quote from the Preface to the English translation of the first edition.

The theory of groups has a long and rich history. Arising from the needs of Galois theory, it developed at first as the theory of finite substitution groups (Cauchy, Jordan, Sylow). However, it was fairly soon discovered that for the majority of problems that are of interest to the theory this special material - namely substitutions - used in the construction of the groups is not essential and that the actual topic is the study of properties of a single algebraic operation defined in a set consisting of a finite number of elements of an arbitrary nature. This discovery, which may appear trivial to-day, turned out to be, in fact, very fruitful and led to the creation of the general theory of finite groups. True, the transition from substitution groups to arbitrary finite groups did not essentially extend the realm of the objects to be studied; however, it put the theory on an axiomatic basis, gave it order and clarity, and thus facilitated its further growth.

The golden age of the theory of finite groups came at the end of the last century and the first decade of the present. During this period the fundamental results of the theory were obtained, the fundamental directions of research were laid down, and the fundamental methods were created; generally, through the work of its principal promoters (Frobenius, Hölder, Burnside, Schur, Miller) the theory of finite groups acquired at this time all the essential features it has at the present day. But later it became clear that the finiteness of a group is a restriction that is too strong and not always natural. It was of particular importance that this restriction very soon led to conflicts with the needs of neighbouring branches of mathematics: in several parts of geometry, the theory of automorphic functions, topology, in all of these one again and again came across algebraic structures similar to groups, but infinite, and so demands were made upon the theory of groups that the theory of finite groups was not in a position to satisfy. Moreover, from the point of view of algebra itself - of which the theory of groups is a part - a situation could hardly be regarded as normal in which such very simple and important groups as, for example, the additive group of integers remained outside the limits of the theory. The finite group must therefore be a special case of the general concept of a group, and the theory of finite groups must be a chapter in the general theory of "infinite" (that is, not necessarily finite) groups.

An exposition of the elements of group theory without the assumption that the groups under consideration were finite was, for the first time in the whole literature, made in the book Abstract Theory of Groups [in Russian] by 0 J Schmidt (Kiev 1916), a book which even now remains a reference work for all Soviet algebraists. But the broader development of the general theory of groups began somewhat later and was linked with that radical reorganization and transition to a set-theoretical foundation in algebra which occurred in the twenties of the present century (Emmy Noether). It was from here that the new concepts of operator systems and chain conditions were introduced into the theory of groups.

Subsequently the work on the general theory of groups became very vigorous and varied, and at the present time this part of mathematics hasbecome a wide and rich science occupying one of the foremost places in contemporary algebra. Clearly this development of the general theory of groups could not ignore the achievements of the theory of finite groups. On the contrary, many results sprang from the corresponding parts of the theory of finite groups; the guiding principle was the endeavour to replace the finiteness of the group by other natural restrictions under which a given theorem or a given theory remain valid but without which they cease to hold. Furthermore, very often a problem that is simple and completely solved in the case of finite groups changes to a broad theory, yet far from complete, this happens, for example, in the theory of abelian groups, one of the most important parts of contemporary group theory. At the same time a number of new branches arose, linked essentially with the study of infinite groups - the theory of free groups and of free products. Finally, in some cases, above all in the problem of giving a group by defining relations, the theory of groups achieved for the first time a clarity and rigour that had been lacking in the preceding stage of its development.

The theory of groups is far from complete. The variety of concrete problems confronting it and the fact that in some directions the research work has only recently begun justify us in assuming that the general theory of groups has not yet passed the climax of its growth. Nevertheless the time has come to systematize the rich material already accumulated and thus to present to a wide circle of mathematicians the basic trends of contemporary group theory, its methods, its principal achievements, and finally, the immediate problems facing it and the paths along which it will necessarily develop in the near future.

The present book does not pretend, obviously, to range over the whole theory of groups; but almost all the main branches of our science are presented in it, to an extent sufficient to show the reader the wealth of its contents and the variety of its methods.

The reader is not required to have a preliminary acquaintance with the elementary concepts of the theory of groups. A basic course of higher algebra is a prerequisite only for some initial examples of groups, such as matrices, permutations, roots of unity. As to the theory of numbers, the reader need only know the elements of the theory of congruences. On the other hand, the reader should be thoroughly acquainted with the elements of the theory of sets, as far as the first four chapters of the book Set Theory by Hausdorff (Chelsea, 1956). In particular, in many constructions and proofs transfinite induction is an essential tool.

The bibliography contains, as far as possible, a complete list of papers on the general theory of groups, including some that have come out recently but have not influenced the book. Of the rich literature on finite groups the bibliography includes only a few directly connected with the contents of the book. References to the bibliography are given in the text by the name of the author and (in brackets) the number of the paper quoted.

Moscow, October 1940
A KUR0SH

Last Updated March 2006