Difference sets disjoint from a subgroup II: groups of order $4p^2$

TL;DR

This paper proves that no DRAD difference sets exist in groups of order 4p^2, where p is an odd prime, expanding understanding of difference sets in finite group theory.

Contribution

It establishes the non-existence of DRAD difference sets in groups of order 4p^2 for odd primes p, filling a gap in the classification of such difference sets.

Findings

No DRAD difference sets in groups of order 4p^2 for odd p

Extends non-existence results in difference set theory

Contributes to classification of difference sets in finite groups

Abstract

We study finite groups $G$ having a normal 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 {\it DRAD difference set}). We show that there are no such groups of order $4p^2$, where $p$ is an odd prime.