Character theoretic techniques for nonabelian partial difference sets

TL;DR

This work develops nonabelian, character-theoretic techniques for (v, k, λ, μ)-partial difference sets, extending Ott's framework to constrain conjugacy-class intersections and derive both nonexistence results and constructive infinite families in nonabelian groups. Central tools include the class function $Φ(h)$, the eigenstructure of the PDS via $θ_1, θ_2$ and Δ, and reductions modulo primes through local rings to obtain strong arithmetic constraints on $|D\cap h^G|$. The authors prove new restrictions, rule out several parameter sets, and construct numerous nonabelian PDSs—some in infinite families tied to block-regular Steiner designs (Clapham, Wilson)—while also providing computational pipelines (modular and non-modular) to search for and verify PDSs. The results expand the landscape of PDSs beyond abelian groups, with implications for SRGs and combinatorial designs, and offer practical methods to generate and certify new examples. These techniques illuminate how group-theoretic structure governs PDS feasibility and enable systematic exploration of otherwise intractable nonabelian cases.

Abstract

A $(v,k,λ, μ)$-partial difference set (PDS) is a subset $D$ of size $k$ of a group $G$ of order $v$ such that every nonidentity element $g$ of $G$ can be expressed in either $λ$ or $μ$ different ways as a product $xy^{-1}$, $x, y \in D$, depending on whether or not $g$ is in $D$. If $D$ is inverse closed and $1 \notin D$, then the Cayley graph ${\rm Cay}(G,D)$ is a $(v,k,λ, μ)$-strongly regular graph (SRG). PDSs have been studied extensively over the years, especially in abelian groups, where techniques from character theory have proven to be particularly effective. Recently, there has been considerable interest in studying PDSs in nonabelian groups, and the purpose of this paper is develop character theoretic techniques that apply in the nonabelian setting. We prove that analogues of character theoretic results of Ott about generalized quadrangles of order $s$ also hold in the general PDS setting, and we are able to use these techniques to compute the intersection of a putative PDS with the conjugacy classes of the parent group in many instances. With these techniques, we are able to prove the nonexistence of PDSs in numerous instances and provide severe restrictions in cases when such PDSs may still exist. Furthermore, we are able to use these techniques constructively, computing several examples of PDSs in nonabelian groups not previously recognized in the literature, including an infinite family of genuinely nonabelian PDSs associated to the block-regular Steiner triple systems originally studied by Clapham and related infinite families of genuinely nonabelian PDSs associated to the block-regular Steiner $2$-designs first studied by Wilson.

Theorems & Definitions (64)

• Lemma 2.1
• Lemma 2.2
• Definition 2.3
• Lemma 2.4
• Theorem 2.5
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3