Let $G$ be a finite group. Let $H \leq G$ such that $H^x \cap H = 1$ for all $x \in G \setminus H$. I wish to show that $H$ is a Hall subgroup of $G$.
Here's what I have so far. Let $|H|=n$ and $|G:H|=m$. So we want to show $\gcd(m,n)=1$. I try to look at $N_G(H)$. $H$ is not normal because otherwise $H^g=H$ for all $g \in G$. So $N_G(H)<G$, a strict inclusion. And of course we also have $H \lhd N_G(H)$, and $|G:H|=|G:N_G(H)||N_G(H):H|$. But I'm unable to deduce much about the indexes on the right hand side of this last equation.
How should I proceed to show that $H$ is a Hall subgroup?