= 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>.
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