The History of Combinatorial Group Theory: A Case Study in the History of Ideas

ISBN-13: 9781461394891 / Angielski / Miękka / 2011 / 234 str.

B. Chandler; W. Magnus

Opis

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."

Spis treści

I The Beginning of Combinatorial Group Theory.- I.I Introduction to Part I.- I.2 The Foundations: Dyck’s Group-Theoretical Studies.- I.3 The Origin: The Theory of Discontinuous Groups.- I.4 Motivation: The Fundamental Groups of Topological Spaces.- I.5 The Graphical Representation of Groups.- I.6 Precursors of Later Developments.- A. Arithmetically Defined Linear Groups in Higher Dimensions.- B. Arithmetically Defined Linear Groups in Two Dimensions.- C. Geometric Constructions. Fuchsian Groups.- D. Braid Groups and Mapping Class Groups.- E. Differential Equations, Linear Groups, and Lie Groups.- F. Finite Groups.- I.7 Summary.- I.8 Modes of Communication. Growth and Distribution of Research in Group Theory.- I.9 Biographical Notes.- I.10 Notes on Terminology and Definitions.- I.11 Sources.- II The Emergence of Combinatorial Group Theory as an Independent Field.- II. 1 Introduction to Part II.- II.2 Free Groups and Their Automorphisms.- II.3 The Reidemeister—Schreier Method.- II.4 Free Products and Free Products with Amalgamations.- II.5 One-Relator Groups.- II.6 Metabelian Groups and Related Topics.- A. The Principal Ideal Theorem.- B. Applications to the Theory of Knots and Links.- C. A Problem from the Foundations of Geometry.- D. Notes on Later Developments and Generalizations.- II.7 Commutator Calculus and the Lower Central Series.- II.8 Varieties of Groups.- II.9 Topological Properties of Groups and Group Extensions.- II.10 Notes on Special Groups.- II.11 Postscript: The Impact of Mathematical Logic.- II.12 Modes of Communication.- II.13 Geographical Distribution of Research and Effects of Migration.- II.14 Organization of Knowledge.- Index of Names.- Index of Subjects.