I'm attempting to prove that the action of the dihedral group, $D_{2n}$, is primitive on the set of vertices of the regular $n$-gon $\iff\,n$ is prime.
I'm not wholly sure where to start. I know that a cyclic group that acts transitively on a set of non-prime order doesn't have a primitive action on that set, and I am trying to formulate a proof based on this fact.
Does anyone have any pointers? Thank you :)