The History of Combinatorial Group Theory pp 11-13 | Cite as

# The Origin: The Theory of Discontinuous Groups

## Abstract

Even apart from Dyck’s own testimony, there exists very strong evidence that the theory of discontinuous groups was the basis for the group theoretical studies of Dyck [1882]. Dyck was a student of F. Klein and, in 1882, was his assistant at the University of Leipzig. Shortly after the publication of Dyck’s paper, there appeared two very important papers in the theory of discontinuous groups by the two leading mathematicians working in this field at that time. One of them was a long paper (61 pages) by H. Poincaré [1882] on fuchsian groups. In the introduction, Poincaré says that he had published sketches of his ideas and results earlier but that he would now try to give a systematic account of the theory. Almost simultaneously, there appeared a long (78 pages) paper by F. Klein [1883] which is particularly important for the theory of group presentations since it contains what is now known as Klein’s theory of “composition of groups.” This term was actually coined by Fricke and Klein [1897, pp. 190–194]. Klein [1883] calls it *Ineinanderschiebung* (“meshing”) of groups (p. 200). The composition of groups can be viewed as a solution of the word problem for the free product of groups which act in a certain manner on the points of a topological space. The most simple example, described by Fricke and Klein, is the following one. Consider four disjoint circular disks *D*_{1}, *D’*_{1}, *D*_{2}, *D’*_{2} in the complex plane.

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