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Dover The Genesis of the Abstract Group Concept A Contribution to the History of the Origin of Abstract Group Theory 2007 Edition by Hans Wussing, Abe Shenitzer, Hardy Grant
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 :- PrefacePreface to the American EditionTranslator's NoteIntroductionPart I. Implicit Group-Theoretic Ways of Thinking in Geometry and Number Theory1. Divergence of the different tendencies inherent in the evolution of geometry during the first half of the nineteenth century2. 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 conceptPart II. Evolution of the Concept of a Group as a Permutation Group1. Discovery of the connection between the theory of solvability of algebraic equations and the theory of permutations2. Perfecting the theory of permutations3. The group-theoretic formulation of the problem of solvability of algebraic equations4. The evolution of the permutation-theoretic group concept5. The theory of permutation groups as an independent and far-reaching area of investigationPart III. Transition of the Concept of a Transformation Group and the Development of the Abstract Group Concept1. The theory of invariants as a classification tool in geometry2. Group-theoretic classification of geometry: The Erlangen Program of 18723. Groups of geometric motions; Classification of transformation groups4. The shaping and axiomatization of the abstract group conceptEpilogueNotesBibliographyName IndexSubject Index