Restrictions on parameters of partial difference sets in nonabelian groups

Abstract

A partial difference set $S$ in a finite group $G$ satisfying $1\notin S$ and $S=S^{-1}$ corresponds to an undirected strongly regular Cayley graph $\mathrm{Cay}(G,S)$. While the case when $G$ is abelian has been thoroughly studied, there are comparatively few results when $G$ is nonabelian. In this paper, we provide restrictions on the parameters of a partial difference set that apply to both abelian and nonabelian groups and are especially effective in groups with a nontrivial center. In particular, these results apply to $p$-groups, and we are able to rule out the existence of partial difference sets in many instances.

corrected typos from first version; to appear in Journal of Combinatorial Designs