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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 { ∈ S × S : x ≠ y } {\displaystyle \{\in S\times S:x\neq y\}}. That is, assuming that G {\displaystyle G} acts on the left of S {\displaystyle S} , for each pair of pairs , ∈ S × S {\displaystyle ,\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 = {\displaystyle g=}.
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 = {\displaystyle g=}.