Multiply transitive group action

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)} .

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