Two-transitive group action

A group action on a set $X$ with at least two points is two-transitive when it acts transitively on ordered pairs of distinct points. Equivalently, the group action is transitive and the stabilizer subgroup of one point acts transitively on the remaining points. First move the initial first point to the target first point, then use that stabilizer subgroup to move the second point. Conversely, two-transitivity supplies both kinds of moves.

The transitivity requirement cannot be omitted: a group that fixes one point and permutes all others has a stabilizer subgroup transitive on the remaining points, while the whole group action is not even transitive. This is the $n=2$ case of an n-transitive group action.

A point stabilizer in a finite two-transitive group action 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.

There are $\left(\genfrac{}{}{0ex}{}{6}{3}\right)/2=10$ partitions of six letters into two unordered triples. A stabilizer is $S_3\wr S_2$, also the normalizer of the Sylow subgroup generated by one 3-cycle in each triple. Relative to one partition, all other partitions have intersection pattern $1+2$ up to interchange, giving one orbit of the stabilizer and hence a two-transitive action. A natural $S_5$ fixing one letter is transitive on these partitions, giving $S_6=S_5(S_3\wr S_2)$.

A minimal normal subgroup of a finite faithful two-transitive group action is either regular elementary abelian or primitive nonabelian simple. The imprimitive case reduces to two-point stabilizers being trivial, then counting derangements gives a characteristic regular Sylow subgroup. In the primitive nonregular case, each simple direct factor is transitive. Two commuting transitive factors are regular and are mutual full centralizers. Thus at most two factors occur and $|N|=n^2$. But the common subdegree of $N_{\alpha }$ then divides both $n$ and $n-1$, forcing $N_{\alpha }=1$, a contradiction. Only one factor remains.

Translations are regular on $V$, and the zero stabilizer is $H$. Two-transitivity is equivalent to $H$ being transitive on nonzero vectors. Full affine linear groups and groups with special linear or symplectic linear parts give examples.