The History of Combinatorial Group Theory: A Case Study in the History of Ideas (Studies in the History of Mathematics and the Physical Sciences)

One of the pervasive phenomena in the history of science is the development of independent disciplines from the solution or attempted solutions of problems in other areas of science. In the Twentieth Century, the creation of specialties witqin the sciences has accelerated to the point where a large number of scientists in any major branch of science cannot understand the work of a colleague in another subdiscipline of his own science. Despite this fragmentation, the development of techniques or solutions of problems in one area very often contribute fundamentally to solutions of problems in a seemingly unrelated field. Therefore, an examination of this phenomenon of the formation of independent disciplines within the sciences would contrib­ ute to the understanding of their evolution in modern times. We believe that in this context the history of combinatorial group theory in the late Nineteenth Century and the Twentieth Century can be used effectively as a case study. It is a reasonably well-defined independent specialty, and yet it is closely related to other mathematical disciplines. The fact that combinatorial group theory has, so far, not been influenced by the practical needs of science and technology makes it possible for us to use combinatorial group theory to exhibit the role of the intellectual aspects of the development of mathematics in a clearcut manner. There are other features of combinatorial group theory which appear to make it a reasona­ ble choice as the object of a historical study. Front Matter....Pages i-viii Front Matter....Pages 1-1 Introduction to Part I....Pages 3-4 The Foundation: Dyck’s Group-Theoretical Studies....Pages 5-10 The Origin: The Theory of Discontinuous Groups....Pages 11-13 Motivation: The Fundamental Groups of Topological Spaces....Pages 14-21 The Graphical Representation of Groups....Pages 22-28 Precursors of Later Developments....Pages 29-50 Summary....Pages 51-57 Modes of Communication. Growth and Distribution of Research in Group Theory....Pages 58-67 Biographical Notes....Pages 68-70 Notes on Terminology and Definitions....Pages 71-74 Sources....Pages 75-76 Front Matter....Pages 77-77 Introduction to Part II....Pages 79-80 Free Groups and Their Automorphisms....Pages 81-90 The Reidemeister—Schreier Method....Pages 91-101 Free Products and Free Products with Amalgamations....Pages 102-112 One-Relator Groups....Pages 113-121 Metabelian Groups and Related Topics....Pages 122-140 Commutator Calculus and the Lower Central Series....Pages 141-156 Varieties of Groups....Pages 157-161 Topological Properties of Groups and Group Extensions....Pages 162-174 Front Matter....Pages 77-77 Notes on Special Groups....Pages 175-180 Postscript: The Impact of Mathematical Logic....Pages 181-186 Modes of Communication....Pages 187-192 Geographical Distribution of Research and Effects of Migration....Pages 193-199 Organization of Knowledge....Pages 200-206 Back Matter....Pages 207-234