This book contains an exposition of those parts of group theory which arise from the presentation of groups in terms of generators and defining relations. Groups appear naturally in this form in certain topological problems, and the first serious contributions to this part of group theory were made by Poincaré, Dehn, Tietze, and other topologists. The name "Combinatorial Group Theory" refers to the frequent occurrence of combinatorial methods, which seem to be characteristic of this discipline.
The book is meant to be used as a textbook for beginning graduate students who are acquainted with the elements of group theory and linear algebra. The first two chapters are of a fairly elementary nature, and a particularly large number of exercises were included in these parts. The exercises are not always easy ones, but the hints given are usually broad enough to make them so. Some interesting results have been presented in the form of exercises; the text proper does not make use of these results except where specifically indicated. (It is a good idea for the reader to examine the exercises even if he does not wish to attempt them.)
There is not very much overlapping of the topics presented here with those treated in the books on group theory by A Kurosh and by Marshall Hall, Jr. The subjects of Nielsen Transformations (Chapter 3), Free and Amalgamated Products (Chapter 4), and Commutator Calculus (Chapter 5) are treated here in a more detailed fashion than in the works of Kurosh and of Hall.
All theorems which are labelled with a number are proved in full. However, we have stated some advanced results without proof, whenever the original proofs were long and could not be amalgamated with the main body of the text. Such results are stated either as theorems labelled with the name of the author (e.g., Grushko's Theorem) or with a letter and number (e.g., Theorems N1 to N13 on Nielsen transformations, or T1 to T5 on topological aspects).
We have tried to give references to relevant papers and monographs in the later parts of the book (after the first two chapters). Usually, such references are collected at the end of each section under the heading "References and Remarks."
The sixth (and last) chapter contains a brief survey of some recent developments. It is hardly necessary to say that we could not even try to give a complete account. We are painfully aware of the many gaps. Some methods and results, as well as references, may have escaped our attention altogether.
Over the years, we have received suggestions and criticisms from many mathematicians, and we owe much to comments from our colleagues as well as from our students. We also wish to acknowledge the help given to us by the National Science Foundation which, through several grants given to New York University and Adelphi University, facilitated the cooperation of the authors.
This book is dedicated to the memory of Max Dehn. We believe this to be more than an acknowledgment of a personal indebtedness by one of the authors who was Dehn's student. The stimulating effect of Dehn's ideas on presentation theory was propagated not only through his publications, but also through talks and personal contacts; it has been much greater than can be documented by his papers. Dehn pointed out the importance of fully invariant subgroups in 1923 in a talk (which was mimeographed and widely circulated but never published). His insistence on the importance of the word problem, which he formulated more than fifty years ago, has by now been vindicated beyond all expectations.
WILHELM MAGNUS, New York University
ABRAHAM KARRASS, Adelphi University
DONALD SOLITAR, Adelphi University
December 1965
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