Graphs, Maps and Cayley Diagrams

Abstract

The chief purpose of this chapter is to describe Cayley’s representation of a group with given generators by a topological 1-complex or graph, whose vertices represent the elements of the group while certain sets of edges are associated with the generators. Cayley (1878a, b) proposed the use of colours to distinguish the edges associated with different generators (see Burnside 1911, pp. 423–427 and the frontispiece). Instead, for the sake of easier printing, we use lines drawn in various styles: ordinary, broken, dotted, etc. After suitably embedding the graph into a surface, we obtain a map from which a set of defining relations for the group may be read off.