On non-symmetric relative difference sets

Abstract

Let D be a (m, u, k, λ)-difference set in a group G relative to a subgroup U of G. We say D is symmetric if D^(−1) is also a (m, u, k, λ)-difference set. By a result of [7] D is symmetric if U is a normal subgroup of G. In general, D is non-symmetric when U is not normal in G. In this paper we study a condition under which D is symmetric and show that if D is semiregular then D is symmetric if and only if the dual of dev(D) is a divisible design. We also give a modification of Davis' product construction of relative difference sets and as an application we give a class of non-symmetric semiregular relative difference sets.