I'm attempting Exercise 3.4.6 in Dixon & Mortimer's book on Permutation Groups:
Let $G$ be a finite primitive permutation group with abelian point stabilisers. Show that $G$ has a regular normal elementary abelian $p$-subgroup for some prime $p$.
I know that if $G$ is a primitive permutation group with abelian point stabilisers and $G$ does not have prime order, then it is a Frobenius Group. Is it also a Frobenius group if $G$ has finite order?
If so, I know that the Frobenius kernel $K$ is a normal regular subgroup from the Structure Theorem for Finite Frobenius Groups. I'm not sure how to go about proving that $K$ is also an elementary abelian $p$-subgroup though.
From the Frattini Argument I know that if $P$ is a Sylow $p$-subgroup of $K$ then the direct product of $K$ and the normaliser of $P$ is equal to $G$. But if $G$ is Frobenius it is also equal to the semi-direct product of the Frobenius kernel and complement - perhaps I can use this to show that $K$ itself is a $p$-subgroup? And therefore an elementary group since a direct product of a $p$-subgroup and the identity (which is finite and cyclic or order $1$).
I'm a bit stuck as to where to start on proving that $K$ is abelian, and this is all assuming that $G$ is a Frobenius group and $K$ is the subgroup in question!
Any help would be much appreciated. Thanks!