Given a group $G$ acting on a set $X$ with $|X| \ge 2$, we say that the action of $G$ on $X$ is 2-transitive iff for any $y_1,y_2 \in X$ with $y_1 \neq y_2$ and $x_1,x_2 \in X$ with $x_1 \neq x_2$ there is $g \in G$ such that $g \cdot x_1 = y_1$ and $g \cdot x_2 = y_2$
The thing that is not too clear from this definition is that I could simply take $x_1 = y_1$ and $x_2 = y_2$ and then $g = \{e\}$ to make this always hold. Is maybe implied in the definition that we should choose a non-trivial $g \in G$?