 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.