Cayley graphs and metric spaces: An investigation of groups and their generators

Claire Ihlendorf and Erin Kelly

A choice of generators for a discrete group determines a Cayley graph, and hence a metric space structure, for the group. This paper investigates the relationship between the group operation and the metric. In particular this paper formulates, and proves for some classes of groups, the following conjecture: for a finite group of even order, generated by two or fewer generators, there exists a Cayley graph that is constructed using no more than two generators and in which the element farthest from the identity has order two. (advisor Peter Haskell)