Restrictions on parameters of partial difference sets in nonabelian groups

TL;DR

This paper establishes new restrictions on the parameters of partial difference sets in finite groups, especially nonabelian ones, aiding in the classification and nonexistence proofs of such sets.

Contribution

It introduces parameter restrictions that apply to both abelian and nonabelian groups, with particular effectiveness in groups with nontrivial centers, including p-groups.

Findings

Restrictions rule out many potential partial difference sets in nonabelian groups.

Results extend known conditions from abelian to nonabelian groups.

Applications include nonexistence proofs for certain p-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 ${\rm Cay}(G,S)$. 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.