Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations

From about the middle of the 19th century, group theory divided itself into two parts: discrete groups and topological groups. Most research about discrete groups centered around finite groups. However, by the end of the century, finitely generated abelian groups had been classified, and von Dyck had introduced free groups and presentations of groups by generators and relations. Inspired by questions about fundamental groups, Dehn posed the word problem: does every finitely presented group possess an algorithm which can determine whether an arbitrary word on its generators is equal to the identity?

For the next fifty years, finite groups continued to be the main focus of research in discrete groups, but some important results about infinite groups were discovered: the Schreier-Neilsen theorem that subgroups of free groups are free, and its generalization by Kurosh that subgroups of free products are themselves free products; the Freiheitssatz of Magnus described the structure of groups with only one defining relation; Ulm classified all countable torsion abelian groups; Baer began the study of torsion-free abelian groups.

The fifteen years after World War II produced, among other results, the study of "small cancellation" theory; M. Hall solved the Burnside problem for exponent 6: a finitely generated group G in which x⁶ = 1 for all x in G must be finite; Novikov and Boone, independently, proved that there exist finitely presented groups having an unsolvable word problem; and Graham Higman, in 1961, characterized subgroups of finitely related groups in terms of recursive functions.

But it is fair to say that most group-theorists in the 1960s focused on finite groups. For example, Suzuki discovered a new family of finite simple groups in 1960, and Feit and Thompson, in 1963, proved that every group of odd order is solvable, thereby solving another old problem of Burnside and contributing mightily to new investigations into finite simple groups. Thus, when Combinatorial Group Theory appeared in 1966, it described a relatively small corner of group theory, though one in which new significant theorems were being proved.

In his review in Math Reviews [MR 0207802 (34#7617)], Higman wrote:

This book is an excellent and detailed account, with many examples, of some aspects of group theory closely connected with generators and relations. The approach throughout is very concrete. Thus in the first chapter the group defined by a presentation is constructed as a group of equivalence classes of words. A free group is then defined as one given by a presentation whose set of relators is empty, and the definition by the universal property is relegated to an exercise. Also discussed in this chapter is the relation between different presentations of the same group. In the second chapter the authors deal with presentations of subgroups, prove the subgroup theorem for free groups by the Reidemeister-Schreier method, and derive Takahasi's theorem and the Hopfian property of finitely generated free groups. The third chapter deals with Nielsen transformations and with some tests for isomorphism. It also contains an account, without proofs, of the automorphisms of finitely generated free groups. The fourth chapter contains an account of free products, perhaps with an amalgmation, the Kurosh subgroup theorem and the theory of groups with a single defining relation. (This last section is one of the best in the book, and contains a tremendous amount of information not readily accessible elsewhere.) The fifth chapter deals with commutator calculus and the lower central series of free groups with applications, for instance, to Burnside's problem. Finally, a short sixth chapter gives brief indications of some recent developments.

As will be seen from this description of the contents, this book covers a section of group theory not adequately treated elsewhere in the literature. The treatment is very thorough, almost everything being written out in explicit detail. If this makes for a heavy book, materially, for a comparatively small corner of mathematics, it must be said also that it makes the book a delight to read, and easy to find one's way around. In spite of the very numerous problems, the reviewer sees this rather as a reference book than as a textbook, but for anyone with any interest in infinite groups, it will be indispensable.

In their preface to the 1976 reprint of Combinatorial Group Theory, the authors wrote:

The original edition of this book was published (...) in 1966. Since then, a large number of important papers in combinatorial group theory have appeared. To embody at least a substantial part of their results in the present edition would have been possible only by increasing the size of the book considerably. Instead, we have confined ourselves to the correction of those errors and misprints of which we are aware...

They go on to mention several books containing updated references and more recent developments. Perhaps the most influential book written afterward is Combinatorial Group Theory, by R. Lyndon and P. Schupp, Springer-Verlag, 1977. We quote from their preface.

A major stimulus to the study of infinite discontinuous groups was the development of topology: we mention particularly the work of Poincaré, Dehn, and Nielsen. This last influence is especially important in the present context since it led naturally to the study of groups presented by generators and relations....

Important contributions to the development of the ideas initiated by Dehn were made by Magnus, who has in turn been one of the strongest influences on contemporary research. The book Combinatorial Group Theory, by Magnus, Karrass, and Solitar, which appeared in 1966 and immediately became the classic in its field, was dedicated to Dehn. It is our admiration for that work which has prompted us to give this book the same title.

Among many important recent results proved since the book's appearance, let us mention only the solution to Burnside's problem: Adjan, Novikov, Ol'shanskii, and Ivanov have shown, for sufficiently large exponent n, that there exist infinite finitely generated groups of exponent n. And Zelmanov has shown that among all finite groups of exponent n having k generators there is one of largest order. A separate essay would be required to describe details of other modern work in combinatorial group theory.

Now that the classification of the finite simple groups, with its many applications, is complete, combinatorial group theory is the pre-eminent branch of (discrete) groups. It is appropriate that the book of Magnus, Karrass, and Solitary be reprinted, for it is a superb introduction to this beautiful and active part of mathematics. But this book is not merely an introduction. The themes, ideas, and methods contained in it are still being developed and used in studying, for example, hyperbolic groups, geometric group theory, and the Bass-Serre theory of groups acting on trees.

Joseph Rotman (rotman@math.uiuc.edu) is Professor Emeritus of Mathematics at the University of Illinois at Urbana-Champaign.