The History of Combinatorial Group Theory

1. Front Matter

   Pages i-viii

   PDF

2. The Beginning of Combinatorial Group Theory

   1. Front Matter

      Pages 1-1

      PDF

   2. Introduction to Part I

      • Bruce Chandler, Wilhelm Magnus

      Pages 3-4

   3. The Foundation: Dyck’s Group-Theoretical Studies

      • Bruce Chandler, Wilhelm Magnus

      Pages 5-10

   4. The Origin: The Theory of Discontinuous Groups

      • Bruce Chandler, Wilhelm Magnus

      Pages 11-13

   5. Motivation: The Fundamental Groups of Topological Spaces

      • Bruce Chandler, Wilhelm Magnus

      Pages 14-21

   6. The Graphical Representation of Groups

      • Bruce Chandler, Wilhelm Magnus

      Pages 22-28

   7. Precursors of Later Developments

      • Bruce Chandler, Wilhelm Magnus

      Pages 29-50

   8. Summary

      • Bruce Chandler, Wilhelm Magnus

      Pages 51-57

   9. Modes of Communication. Growth and Distribution of Research in Group Theory

      • Bruce Chandler, Wilhelm Magnus

      Pages 58-67

   10. Biographical Notes

       • Bruce Chandler, Wilhelm Magnus

       Pages 68-70

   11. Notes on Terminology and Definitions

       • Bruce Chandler, Wilhelm Magnus

       Pages 71-74

   12. Sources

       • Bruce Chandler, Wilhelm Magnus

       Pages 75-76

3. The Emergence of Combinatorial Group Theory as an Independent Field

   1. Front Matter

      Pages 77-77

      PDF

   2. Introduction to Part II

      • Bruce Chandler, Wilhelm Magnus

      Pages 79-80

   3. Free Groups and Their Automorphisms

      • Bruce Chandler, Wilhelm Magnus

      Pages 81-90

   4. The Reidemeister—Schreier Method

      • Bruce Chandler, Wilhelm Magnus

      Pages 91-101

   5. Free Products and Free Products with Amalgamations

      • Bruce Chandler, Wilhelm Magnus

      Pages 102-112

   6. One-Relator Groups

      • Bruce Chandler, Wilhelm Magnus

      Pages 113-121

   7. Metabelian Groups and Related Topics

      • Bruce Chandler, Wilhelm Magnus

      Pages 122-140

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.

• Abelian group
• Definierende Relation
• Erzeugendensystem
• Group theory
• Gruppe (Math.)
• History
• Präsentation (Math.)

• The College of Staten Island of The City University of New York, Staten Island, USA

  Bruce Chandler

• New Rochelle, USA

  Wilhelm Magnus

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