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

The group action is sharply 2-transitive if such g ∈ G {\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, g x = w {\displaystyle gx=w} and g y = z {\displaystyle gy=z} , since the induced action on the distinct set of pairs is g ( x , y ) = ( g x , g y ) {\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, ... ak and b1, ... bk with the property that all the ai are distinct and all the bi are distinct, there is a group element g in G which maps ai to bi for each i between 1 and k. The Mathieu groups are important examples.