Assume a group $G$ acts transitively on a set $\Omega$ and that the size of $\Omega$ is a prime number. Let $H$ be a normal subgroup of $G$ that acts non-trivially on $\Omega$.

Show that any two $H$-orbits of $\Omega$ have the same size. Deduce that the action of $H$ on $\Omega$ is transitive.

Let $\alpha \in \Omega$ and let $G_{\alpha}$ denote the stabiliser of $\alpha$ in $G$. Show that if $H \cap G_{\alpha}$ is trivial, then there is a bijection $\theta: H \rightarrow \Omega$ under which the action of $G_{\alpha}$ on $H$ by conjugation corresponds to the action of $G_{\alpha}$ on $\Omega$.