Normal subgroup orbits in a faithful prime-degree action
= Normal subgroup orbits in a faithful prime-degree action
If a group acts transitively on a set of prime size and $N$ is normal, then the $N$-orbits form a $G$-invariant block system of equal size. They are therefore either singletons or the whole set. In a faithful action the singleton case forces $N=1$, so every nontrivial normal subgroup is transitive.