Let $G$ be a ﬁnite group, and let $\Omega$ be a transitive $G$-space. Assume 1 $\neq H \unlhd G$ and that |$\Omega$| = $p$ where $p$ is prime, and $G \leq Sym(\Omega)$.
Deduce that then $H$ must act transitively on $\Omega$.
So far I have been able to deduce that $p$ divides the order of $G$ as the action is transitive, and the intersection of all stabilizers must be trivial as the action is faithful. However I'm not sure how to deduce that $H$ acts transitively.