The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory(Dover Books on Mathematics)

This informative survey chronicles the process of abstraction that ultimately led to the axiomatic formulation of the abstract notion of group. Hans Wussing, former Director of the Karl Sudhoff Institute for the History of Medicine and Science at Leipzig University, contradicts the conventional thinking that the roots of the abstract notion of group lie strictly in the theory of algebraic equations. Wussing declares their presence in the geometry and number theory of the late eighteenth and early nineteenth centuries.This survey ranges from the works of Lagrange via Cauchy, Abel, and Galois to those of Serret and Camille Jordan. It then turns to Cayley, to Felix Klein's Erlangen Program, and to Sophus Lie, concluding with a sketch of the state of group theory circa 1920, when the axiom systems of Webber were formalized and investigated in their own right.""It is a pleasure to turn to Wussing's book, a sound presentation of history,"" observed the Bulletin of the American Mathematical Society, noting that ""Wussing always gives enough detail to let us understand what each author was doing, and the book could almost serve as a sampler of nineteenth-century algebra. The bibliography is extremely good, and the prose is sometimes pleasantly epigrammatic.""

Table of Contents:
Preface Preface to the American Edition Translator's Note Introduction 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 Epilogue Notes Bibliography Name Index Subject Index

Review :
"It is a pleasure to turn to Wussing's book, a sound presentation of history .... The topic of the book is such that less is said about group theory itself than about the subjects in which it grew. These discussions are far from perfunctory; the 13 pages on Galois, for instance, are an excellent study of the spirit of his work. Wussing always gives enough detail to let us understand what each author was doing, and the book could almost serve as a sampler of 19th-century algebra. The bibliography is extremely good, and the prose is sometimes pleasantly epigrammatic."- William C. Waterhouse, "Bulletin of the American Mathematical Society" (review of the German edition)