Multiply transitive group action

A group ${\displaystyle G}$ acts 2-transitively on a set ${\displaystyle S}$ if it acts transitively on the set of distinct ordered pairs ${\displaystyle \{(x,y)\in S\times S:x\neq y\}}$. That is, assuming (without a real loss of generality) that ${\displaystyle G}$ acts on the left of ${\displaystyle S}$, for each pair of pairs ${\displaystyle (x,y),(w,z)\in S\times S}$ with ${\displaystyle x\neq y}$ and ${\displaystyle w\neq z}$, there exists a ${\displaystyle g\in G}$ such that ${\displaystyle g(x,y)=(w,z)}$.

The group action is sharply 2-transitive if such ${\displaystyle g\in G}$ is unique.

A 2-transitive group is a group such that there exists a group action that's 2-transitive and faithful. Similarly we can define sharply 2-transitive group.

Equivalently, ${\displaystyle gx=w}$ and ${\displaystyle gy=z}$, since the induced action on the distinct set of pairs is ${\displaystyle g(x,y)=(gx,gy)}$.

The definition works in general with k replacing 2. Such multiply transitive permutation groups can be defined for any natural number k. Specifically, a permutation group G acting on n points is k-transitive if, given two sets of points a₁, ... a_k and b₁, ... b_k with the property that all the a_i are distinct and all the b_i are distinct, there is a group element g in G which maps a_i to b_i for each i between 1 and k. The Mathieu groups are important examples.