I am trying to solve the following exercise from Dixon and Mortimer:
Let $G$ be a finite primitive permutation group with abelian point stabilizers. Show that $G$ has a regular normal elementary abelian subgroup.
As hints the authors suggest to use Frobenius' theorem and the Frattini argument.
My attempts: Let $H$ be some of those abelian stabilizers. By primitivity $H$ is maximal. If there is some element $g \in G \setminus H$ with $H \cap H^g \neq 1$ it is easy to show that this intersection contains such a regular normal subgroup we are looking for. So let us suppose there is no such element, i.e. $G$ is a Frobenius group with Frobenius kernel $K$, say. It is easy to see that $K$ is regular and characteristically simple, hence a direct product of isomorphic simple groups. So it remains to show that $K$ is abelian.
Of course by a well known theorem we know that Frobenius kernels are nilpotent in general, and hence $K$ must be abelian in this case. But do we really need such a strong theorem here or is there an easier argumentation? I did not use the Frattini argument yet.