# Difference sets disjoint from a subgroup

## TL;DR

This paper investigates the structure of finite groups with a subgroup and a difference set disjoint from it, revealing normality conditions, size relations, and providing explicit examples and algebraic structures.

## Contribution

It establishes that such groups must have a normal subgroup with order squared, characterizes the difference set distribution, and constructs a family of examples with specific algebraic properties.

## Key findings

- H is normal in G
- |G|=|H|^2
- Existence of a 2-parameter family of examples

## Abstract

We study finite groups $G$ having a subgroup $H$ and $D \subset G \setminus H$ 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 {\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.