Restrictions on parameters of partial difference sets in nonabelian groups

Abstract

A partial difference set S in a finite group G satisfying ∉ S 1 and S S = −1 corresponds to an undirected strongly regular Cayley graph G S Cay(,). 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. K E Y W O R D S partial difference set, strongly regular Cayley graph −1. If the set S is a v k λ μ (, , ,)-PDS, then the Cayley graph G S Cay(,) is a v k λ μ (, , ,)-strongly regular graph (SRG) [17, Proposition 1.1], which means that G S Cay(,) has v vertices, G S Cay(,) is regular of degree k, any two adjacent vertices in G S Cay(,) have exactly λ common neighbors, and any two nonadjacent vertices in G S Cay(,) have exactly How to cite this article: Swartz E, Tauscheck G. Restrictions on parameters of partial difference sets in nonabelian groups.

Genuinely nonabelian partial difference sets

“…Yoshiara [46] used Benson's lemma along with clever counting and group theoretical arguments to prove that no GQ of order $$({t}^{2},t)$$, where $$t\geqslant 2$$, has a group of automorphisms acting regularly on its point set. Inspired by these results, Swartz and Tauscheck [42] proved corresponding results that apply to PDSs. We state an equivalent result here based on the notation of this paper; in what follows, $$\text{Cl}(x)$$ denotes the conjugacy class of the group element $$x\in G$$ and $${C}_{G}(x)$$ denotes the centralizer of $$x$$ in $$G$$.…”

“…Proof of (ii). We include a proof of (ii), since it is not stated explicitly in [42]. We assume without loss of generality that Δ does not divide μ θ θ − ( + 1) 2 1…”

“…Proof of We include a proof of (ii), since it is not stated explicitly in [42]. We assume without loss of generality that $$\sqrt{{\rm{\Delta }}}$$ does not divide $$\mu -{\theta }_{2}({\theta }_{1}+1)$$.…”