Restrictions on parameters of partial difference sets in nonabelian groups

Abstract

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.

Results on partial geometries with an abelian Singer group of rigid type

“…They are first mentioned in the context of abelian Singer groups in [13] and a more detailed study appeared in [4] and [11]. Partial geometries with nonabelian Singer groups are discussed in [17]. In this paper we obtain further evidence that a certain type of partial geometries with an abelian Singer group is very rare, and we formulate a conjecture on this type of geometries.…”