Difference sets disjoint from a subgroup

Abstract

We study finite groups $G$ having a subgroup $H$ and $D\subset G\setminus H$ such that the multiset $\{x{y}^{-1}:x,y\in D\}$ has every non-identity element occur the same number of times (such a $D$ is called a {\it difference set}). We show that $H$ has to be normal, 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 prime index, 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.