Difference Sets Disjoint from a Subgroup

Abstract

We study finite groups G having a non-trivial, proper subgroup H and $$D \subset G {\setminus } H, D \cap D^{-1}=\emptyset ,$$ such that the multiset $$\{ xy^{-1}:x,y \in D\}$$ has every non-identity element occur the same number of times (such a D is called a difference set). We show that $$|G|=|H|^2$$ , and that $$|D \cap Hg|=|H|/2$$ for all $$g \notin H$$ . We show that H is contained in every normal subgroup of index 2, and other properties. We give a 2-parameter family of examples of such groups. We show that such groups have Schur rings with four principal sets, and that, further, these difference sets determine DRADs.