= Solution
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>, $|G|=n|\operatorname{Stab}_G(\omega_1)|$. Applying the same theorem inside the <stabilizer> to its <orbit> of size $n-1$ shows that $n-1$ divides the <stabilizer> order. Hence
$$
\boxed{n(n-1)\mid|G|}.
$$
In the action on ordered pairs there are exactly \b[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.
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