Hans Wussing's classic on The Genesis of the Abstract Group Concept was first published in German in 1969. An English translation appeared in 1984 from MIT Press. The publication of this unaltered reprint of the 1984 edition is great news for anyone interested in the history of modern algebra, and in particular in the evolution of group theory in the 19th century.
Wussing's goal is to trace the evolution of the concept of a group, and especially of an abstract group. One of his main theses, as put by B. H. Neumann in Mathematical Reviews, is that "the roots of the abstract notion of group do not lie, as frequently assumed, only in the theory of algebraic equations, but that they are also to be found in the geometry and the theory of numbers of the end of the 18th and the first half of the 19th centuries." That is hardly controversial among historians nowadays, largely due to the impact of this book.
Wussing begins with implicit uses of groups in geometry and in number theory, then goes on to discuss the birth of the theory of permutation groups in the context of the theory of equations. After following the progress of this theory through Jordan's famous Théorie des Substituitions, he moves on to discuss invariant theory, the Erlangen program and the discovery of (infinite) groups of transformations. Only at the end of the nineteenth century (and at the end of the book) does the notion of an abstract group emerge and take hold.
Wussing is, of course, aware of Cayley's 1854 papers discussing abstract groups. As he points out, however, those papers had little impact. "The conditions favoring appreciation of the abstract, that is formal, approach had not yet fully developed. Also, as long as permutation groups were the only groups under investigation, there was no interest in generalizing the group concept, nor reason to do so." This is an important point, amply supported by the evidence Wussing brings together. Once both permutation groups and infinite groups of transformations (essentially Lie groups) are on the stage, the move towards abstract groups starts to happen.
Wussing's study is convincing and thorough, and the quality of the bibliography has been often noted by reviewers. Of course, much research has happened since. It is alarming, for example, to realize that group representation theory appears in the index only once. It seems clear, forty years after Wussing wrote, that the birth of representation theory also played an important role in moving towards the abstract group concept.
It is a tribute to Wussing's scholarship that very few historians have felt the need to revisit this territory. The recent collection of essays on Episodes in the History of Modern Algebra (1800-1950) , for example, contains nothing on the evolution of group theory in the nineteenth century.
In summary, this is an essential book, and having it back in print (and at a reasonable price) is something to be grateful for.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME, and is the editor of MAA Reviews.
Preface to the American Edition
Part I. Implicit Group-Theoretic Ways of Thinking in Geometry and Number Theory
1. Divergence of the different tendencies inherent in the evolution of geometry during the first half of the nineteenth century
2. The search for ordering principles in geometry through the study of geometric relations (geometrische Verwandtschaften)
3. Implicit group theory in the domain of number theory: The theory of forms and the first axiomatization of the implicit group concept
Part II. Evolution of the Concept of a Group as a Permutation Group
1. Discovery of the connection between the theory of solvability of algebraic equations and the theory of permutations
2. Perfecting the theory of permutations
3. The group-theoretic formulation of the problem of solvability of algebraic equations
4. The evolution of the permutation-theoretic group concept
5. The theory of permutation groups as an independent and far-reaching area of investigation
Part III. Transition of the Concept of a Transformation Group and the Development of the Abstract Group Concept
1. The theory of invariants as a classification tool in geometry
2. Group-theoretic classification of geometry: The Erlangen Program of 1872
3. Groups of geometric motions; Classification of transformation groups
4. The shaping and axiomatization of the abstract group concept