Investigations published within the last fifteen years have greatly deepened our knowledge of groups and have given wide scope to group-theoretic methods. As a result, what were isolated and separate insights before, now begin to fit into a unified, if not yet final, pattern. I have set myself the task of making this pattern apparent to the reader, and of showing him, as well, in the group-theoretic methods, a useful tool for the solution of mathematical and physical problems.
It was a course by E. Artin, given in Hamburg during the Winter Semester of 1933 and the Spring Semester of 1934, which started me on an intensive study of group theory. In this course, the problems of the theory of finite groups were transformed into problems of general mathematical interest. While any question concerning a single object [e.g., finite group] may be answered in a finite number of steps, it is the goal of research to divide the infinity of objects under investigation into classes of types with similar structure.
The idea of O. Hölder for solving this problem was later made a general principle of investigation in algebra by E. Nöther. We are referring to the consistent application of the concept of homomorphic mapping. With such mappings one views the objects, so to speak, through the wrong end of a telescope. These mappings, applied to finite groups, give rise to the concepts of normal subgroup and of factor group. Repeated application of the process of diminution yields the composition series, whose factor groups are the finite simple groups. These are, accordingly, the bricks of which every finite group is built. How to build is indicated—in principle at least—by Schreier’s extension theory. The Jordan-Hölder-Schreier theorem tells us that the type and the number of bricks is independent of the diminution process. The determination of all finite simple groups is still the main unsolved problem.
After an exposition of the fundamental concepts of group theory in Chapter I, the program calls for a detailed investigation of the concept of homomorphic mapping, which is carried out in Chapter II. Next, Chapter III takes up the question of how groups are put together from their simple components. According to a conjecture of Artin, insight into the nature of simple groups must depend on further research on p-groups. The elements of the theory of p-groups are expounded in Chapter IV. Finally, Chapter V describes a method by which solvable factor groups may be split off from a finite group.
For the concepts and methods presented in Chapter II, particularly those in § 7, one may also consult v. d. Waerden, Moderne Algebra I (Berlin 1937). [English translation: Modern Algebra, New York, 1949]. The first part of Chapter III follows a paper by Fitting, while the proof of the basis theorem for abelian groups, and Schreier’s extension theory, are developed on the basis of a course by Artin. The presentation of the theory of p-groups makes use of a paper by P. Hall. The section on monomial representations and transfers into a subgroup has also been worked out on the basis of a course by Artin. In addition one should consult the bibliography at the end of the book.
Many of the proofs in the text are shorter and—I hope—more transparent than the usual, older, ones. The proof of the Jordan-Hölder-Schreier theorem, as well as the proofs in Chapter IV, §§ 1 and 6, owe their final form to suggestions of E. Witt.
I am grateful to Messrs. Brandt, Fitting, Koethe, Magnus, Speiser, Threlfall and v. d. Waerden for their valuable suggestions in reading the manuscript. I also wish to thank Messrs. Hannink and Koluschnin for their help.
The group-theoretic concepts taken up in this book are developed from the beginning. The knowledge required for the examples and applications corresponds to the contents of, say, the book by Schreier and Sperner, Analytische Geometrie und Algebra, Part I (Leipzig, 1935) [English translation: Introduction to Modern Algebra and Matrix Theory, Chelsea Publishing Company, New York, 1951]. A historical introduction to group theory may be found in the book by Speiser, Theorie der Gruppen von endlicher Ordnung (Berlin, 1937).
I would suggest to the beginner that he familiarize himself first with Chapters I and II, Chapter III, §§ 1, 3, 4, 6, and 7, and Chapter IV, §§ 1 and 3, and also with the corresponding exercises. Then the program outlined in this preface will become clear to him.
HANS ZASSENHAUS