On subsets of partial difference sets

Abstract

Let G be a finite group of order v. A k-element subset D of G is called a ( v, k, λ, μ)-partial difference set in G if the expressions gh −1, for g and h in D with g ≠ h, represent each nonidentity element contained in D exactly λ times and represent each nonidentity element not contained in D exactly μ times. Suppose G is abelian and H is a subgroup of G such that gcd (|H|,|G|/|H|) = 1 and |G|/|H| is odd. In this paper, we show that if D is a partial difference set in G with { d −1| d∈ D} = D, then D ∩ H is a partial difference set in H.