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.
A partial difference set S in a finite group G satisfying 1 / ∈ S and S = S −1 corresponds to an undirected Cayley graph 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.
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