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Khinchin. (0-486-60434-9) Arithmetic Refresher, A. Albert Klaf. (0-486-21241-6) Calculus Refresher, A. Albert Klaf. (0-486-20370*0) (continued on back flap) The Genesis of the Abstract Group Concept The Genesis of the Abstract Group Concept A Contribution to the History of the Origin of Abstract Group Theory Hans Wussing Translated by Abe Shenitzer with the editorial assistance of Hardy Grant DOVER PUBLICATIONS, INC. Mineola, New York Copyright Copyright © 1984 by the Massachusetts Institute of Technology All rights reserved. Bibliographical Note This Dover edition, first published in 2007, is an unabridged republication of the work originally published in 1984 by The MIT Press, Cambridge, Massachusetts, It is published by special arrangement with The MIT Press, 55 Hayward Street, Cambridge, MA 02142. Originally published in German in the German Democratic Republic under the title Die Genesis des abstrakten Gruppenbegriffes, © 1969 by VEB Deutscher Verlag der Wissenschaften, Berlin. International Standard Book Number: 0-486-45868-7 Manufactured in the United States of America Dover Publications, Inc., 31 East 2nd Street, Mîneola, N,Y. 11501 Contents Preface 9 Preface to the American Edition И Translator’s Note 13 Introduction 15 Fart 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: The new view of the nature of geometry 25 1 Features of the development of geometry in the beginning of the nineteenth century 25 2 The rise of projective geometry 27 3 Extension of the coordinate concept 28 4 Noneuclidean geometry and the epistemological problem of space 30 5 The «-dimensional manifold as space 32 6 The new conceptual content of “geometry” 33 2 The search for ordering principles in geometry through the study of geometric relations (geometrische Verwandtschaften) 35 1 The study of geometric relations 35 2 Möbius’s classification efforts in geometry 35 3 Plücker’s line geometry 43 4 The role of synthetic geometry in the development of group theory 45 3 Implicit group theory in the domain of number theory: The theory of forms and the first axiomatization of the implicit group concept 48 6 Contents 1 Euler’s paper oa the theory of power residues 48 2 Implicit group theory in Gauss’s work 52 3 Gauss’s theory of composition of forms 55 4 Kronecker’s axiomatization of the implicit group concept 61 Fart П 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 71 1 Lagrange and the theory of solvability of algebraic equations 71 2 Vandermonde and the theory of solvability of algebraic equations 75 3 Beginnings of a group-theoretic treatment of algebraic equations in the work of Lagrange 77 4 Ruffmi’s proof of the unsolvability of the general quintic 80 2 Perfecting the theory of permutations 85 1 The systematic development of permutation theory by Cauchy 86 2 History of the fundamental concepts of the theory of permutations 94 3 The group-theoretic formulation of the problem of solvability of algebraic equations 96 1 Abel and the theory of solvability of algebraic equations 96 2 Galois’s group-theoretic formulation of the problem of solvability of algebraic equations 102 4 The evolution of the permutation-theoretic group concept 118 1 Liouville’s 1846 edition of the principal writings of Galois 118 2 The influence of Galois in Germany, Italy, and England 119 3 The development in France 128 4 Jordan’s commentaries on Galois 135 5 Jordan’s Traité des substitutions et des équations algébriques 141 5 The theory of permutation groups as an independent and far-reaching area of investigation 145 Contents 7 1 Group theory as an independent area of investigation in the work of Serret 145 2 New aspects of the conceptual content of permutation groups in the work of Serret and Jordan 148 Part Ш Transition to 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 167 1 The conceptual background of algebraic invariant theory 167 2 Cayley and the theory of quintics 168 3 The Cayley metric and the determination of the relation between metric geometry and projective geometry 171 4 The Cayley metric as the historical root of the Erlangen Program 175 2 Group-theoretic classification of geometry: The Erlangen Program of 1872 178 1 The intellectual roots of the Erlangen Program 178 2 Klein’s gradual development of the Erlangen Program 182 3 The group-theoretic content of the Erlangen Program 187 3 Groups of geometric motions; Classification of transformation groups 194 1 Motion geometry and groups of geometric motions 194 2 Differentiation and scope of the concept of a discontinuous transformation group 205 3 Lie and the differentiation of the concept of a continuous transformation group: Classification of transformation groups 213 4 Lie and the evolution of the group axioms for infinite groups 223 4 The shaping and axiomatization of the abstract group concept 230 1 The group as a system of defining relations on abstract elements 230 2 The evolution of the abstract group concept 233 3 The group as a fundamental concept of algebra 245 8 Contents 4 The first monographs on abstract group theory 251 5 Group theory around 1920 253 Epilogue 255 Notes 261 Bibliography 295 Name Index 325 Subject Index 329