Nonexistence of Hadamard difference sets in groups of order $4p^2$ with $p\equiv 3 \pmod 4$, $p\ge 7$

Abstract

We show that the Frobenius group $\langle x,y,z\mid x^p=y^p=z^4=1,\ xy=yx,\ zxz^{-1}=y,\ zyz^{-1}=x^{-1}\rangle$ of order $4p^2$ contains no $(4p^2,2p^2-p,p^2-p)$ difference set if $p\equiv 3\pmod 4$ is a prime with $p \geq 7$. Together with results of McFarland, Iiams and Liebler this shows that no group of order $4p^2$ with $p \equiv 3 \pmod 4$ and $p \geq 7$ contains a Hadamard difference set.