an example of a $2$-transitive group

Question

I have a problem to understand the concept of a 2-transitive group.

a group $G$ is transitive if it's action on a set $X$ be transitive, i.e if

$ \forall x,y \in X $ there exists a $ g \in G $ such that $gx=y$.

for example any $k-cyclic$ group acts transitive on a set $[k]$.

but I cant understand what is a $2$-transitive group. could any one give me an example of a $2$-transitive group? here is it's definition :

$ G $ is a $2$-transitive group if it acts on the left of $ S $ in such a way that for each pair of pairs $ \lbrace (x,y),(w,z)\in S\times S\rbrace $ with $x\neq y$ and $w\neq z $ there exists a $g\in G$ such that $ g(x,y)=(w,z)$, Equivalently, $ gx=w$ and $ gy=z$.

I just need an example to understand this concept. please give me an example!

Answer 1

An equivalent definition: $G$ acts doubly transitively on $X$ if the action is transitive, and for any $x \in X$, the stabilizer $G_{x}$ acts transitively on $X \setminus \{x\}$. The symmetric group $S_{n}$ is an example of this.

This is equivalent to having $G$ act on $X \times X$ with two orbits: $\mathcal{O}_{1} = \{ (x,x) : x \in X \}$ and $\mathcal{O}_{2} = \{ (x,y) : x,y \in X, x \neq y\}$.

Answer 2

One example is the group of affine transformations of the plane acting on the plane: Given two ordered pairs of distinct points in the plane, there exists an affine transformation mapping the one pair to the other.

Phrased more concretely; given two distinct points in the $xy$-plane, we can translate, rotate and stretch the plane so that they end up at the origin and $(0,1)$.

Answer 3

The OP perhaps wants a finite 2-transitive group of minimal size. The full symmetric group acting on $\{a,b,c\}$ is an example. Given $x_0\neq y_1$ there are $3\cdot 2=6$ ways to pick $y_0 \neq y_1$ and exactly one permutation that sends $x_i \rightarrow y_i.$

If we take $S=\{0,1,2\}$ to be the integers mod $3$ then we could identify these $6$ permutations with the maps $x\rightarrow ax+b$ for $a=1,2$ and $b=0,1,2$.

That same construction works for any prime $p$.