Abelian Difference Sets Without Self-conjugacy

Abstract

We obtain some results that are useful to the study of abelian difference sets and relative difference sets in cases where the self-conjugacy assumption does not hold. As applications we investigate McFarland difference sets, which have parameters of the form v = q d+1 (q d + q d-1 + ⋯ + q + 2), k = q d (q d + q d-1 + ⋯ + q + 1), λ = q d (q d-1 + q d-2 + ⋯ + q + 1), where q is a prime power and d a positive integer. Using our results, we characterize those abelian groups that admit a McFarland difference set of order k - λ = 81. We show that the Sylow 3-subgroup of the underlying abelian group must be elementary abelian. Our results fill two missing entries in Kopilovich's table with answer "no".