Nonabelian partial difference sets constructed using abelian techniques

TL;DR

This paper extends the construction of partial difference sets from abelian to nonabelian groups, providing the first known examples in certain nonabelian groups and introducing Paley-type PDSs and Paley-Hadamard difference sets in nonabelian contexts.

Contribution

It demonstrates how abelian techniques can be adapted to construct nonabelian partial difference sets, including the first examples of Paley-type PDSs and Paley-Hadamard difference sets in nonabelian groups.

Findings

Constructed nonabelian PDSs of order q^{2m} for odd prime powers q

First known examples of Paley-type PDSs in nonabelian groups

First examples of Paley-Hadamard difference sets in nonabelian groups

Abstract

A $(v,k,\lambda, \mu)$-partial difference set (PDS) is a subset $D$ of a group $G$ such that $|G| = v$, $|D| = k$, and every nonidentity element $x$ of $G$ can be written in either $\lambda$ or $\mu$ different ways as a product $gh^{-1}$, depending on whether or not $x$ is in $D$. Assuming the identity is not in $D$ and $D$ is inverse-closed, the corresponding Cayley graph ${\rm Cay}(G,D)$ will be strongly regular. Partial difference sets have been the subject of significant study, especially in abelian groups, but relatively little is known about PDSs in nonabelian groups. While many techniques useful for abelian groups fail to translate to a nonabelian setting, the purpose of this paper is to show that examples and constructions using abelian groups can be modified to generate several examples in nonabelian groups. In particular, in this paper we use such techniques to construct the first known examples of PDSs in nonabelian groups of order $q^{2m}$, where $q$ is a power of an odd prime $p$ and $m \ge 2$. The groups constructed can have exponent as small as $p$ or as large as $p^r$ in a group of order $p^{2r}$. Furthermore, we construct what we believe are the first known Paley-type PDSs in nonabelian groups and what we believe are the first examples of Paley-Hadamard difference sets in nonabelian groups, and, using analogues of product theorems for abelian groups, we obtain several examples of each. We conclude the paper with several possible future research directions.