Multiply transitive group action

Summary

A group action is $k$-transitive when one group element can move any $k$ distinct points onto any other $k$ distinct points, in order. For $k = 1$ this is a transitive action, and larger $k$ means a stronger kind of symmetry. The symmetric group $S_n$ is $n$-transitive and the alternating group $A_n$ is $(n-2)$-transitive on $n$ points, and a $2$-transitive group is always primitive. Apart from symmetric and alternating groups, finite multiply transitive groups are rare: the Mathieu groups are famous examples.