 Elementary abelian regular kernels in doubly transitive groups

A point stabilizer in a doubly transitive group acts transitively by conjugation on the nonidentity elements of a regular normal subgroup $N$. All those elements have the same order; Cauchy's theorem makes this a prime p and excludes other primes from $|N|$. A finite p-group has a nontrivial characteristic center. Transitivity then forces all of $N$ into its center, so $N$ is elementary abelian. Choosing a vector-space basis identifies the original permutation group with a subgroup of the corresponding general affine group.