The Graphical Representation of Groups

Abstract

The title of this chapter is part of the heading of Chapter 19 of Burnside’s Theory of Groups of Finite Order, 2nd edition, which appeared in 1911. It comprises the description of a group by a tessellation of the sphere, the euclidean or the non-euclidean plane by replicas of a fundamental region of the group which, in this case, must act discontinuously on the manifold in question and also the one-dimensional complex which can be associated with any group presentation and is known under the names “Cayley diagram,” “colour group,” Gruppenbild, and “graph of a group.” We shall use the last term here.