Past exam of the options I course of the University of Oxford / 2008 / A (Groups in Action)

Fix ${\omega }_1$. For any $u,v\ne {\omega }_1$, apply the given two-point transitivity property to $({\omega }_1,u)$ and $({\omega }_1,v)$. The resulting element fixes ${\omega }_1$ and sends $u$ to $v$, so the stabilizer is transitive on the complement.

The whole group is transitive on $\Omega$: extend any desired first-coordinate pair by choosing a different second point on each side. By the orbit-stabilizer theorem, $|G|=n|{\mathrm{Stab}}_G({\omega }_1)|$. Applying the same theorem inside the stabilizer to its orbit of size $n-1$ shows that $n-1$ divides the stabilizer order. HenceIn the action on ordered pairs there are exactly two orbits. One is the diagonal, transitive because the action on $\Omega$ is transitive. The other comprises distinct-coordinate pairs and is transitive by hypothesis. Equality of the two coordinates is preserved by every group element, so the two sets cannot merge into one orbit. Faithfulness of the action is not required.