Ambiguous Groups and Cayley Graphs—A Problem in Distinguishing Opposites

Abstract

SummaryFor any finite group, if we are given the complete Cayley graph for the group, and if the undirected edges are colored in a natural way, does this partial information determine the multiplication table for the group? It turns out that that the answer to this inverse problem is usually yes, but not always. We call a group whose multiplication table cannot be determined from its complete colored Cayley graph an ambiguous group. A simple example of such a group is the quaternion group. We are able classify all ambiguous groups. We show that the complete Cayley graph with colored edges does determine the isomorphism class for the group. Along the way we revisit contributions made to the development of group theory by the eminent mathematicians Cayley, Hamilton, Dirichlet, and Baer.