Doubly transitive permutation group

A 2-transitive group is a transitive group used in group theory in which the stabilizer subgroup of every point acts transitively on the remaining points. Equivalently, 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 ${\displaystyle g(x,y)=(w,z)}$. Equivalently, ${\displaystyle gx=w}$ and ${\displaystyle gy=z}$, since the induced action on the distinct set of pairs is ${\displaystyle g(x,y)=(gx,gy)}$.

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