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

We study finite groups G having a subgroup H and D ⊂ G ∖ H such that (i) the multiset { xy^-1:x,y ∈ 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 ∩ D^(-1) =∅ . We show that |H|=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.