Difference Sets Disjoint from a Subgroup III: The Skew Relative Cases

We study finite groups G having a subgroup H and D subset of G\H such that (i) the multiset {xy(-1) : x, y is an element of D} has every element that is not in H occur the same number of times (such a D is called a relative difference set); (ii) G = D. D(-1). H; (iii) D boolean AND D(-1) = empty set. We show that vertical bar H vertical bar = 2, that H is central and that G is a group with a single involution. We also show that G cannot be abelian. We give infinitely many examples of such groups, including certain dicyclic groups, by using results of Schmidt and Ito.