Lemma 43.23.5. Let $V$ be a vector space. Let $G = \text{PGL}(V)$. Then $G \times \mathbf{P}(V) \to \mathbf{P}(V)$ is doubly transitive.

**Proof.**
Omitted. Hint: This follows from the fact that $\text{GL}(V)$ acts doubly transitive on pairs of linearly independent vectors.
$\square$

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