Sharply Transitive

Definition

$G$ is said to be sharply $n$-transitive on $\Omega$ if for any $({\omega }_1,\cdots ,{\omega }_k)$ and $({\omega }_1^{\prime },\cdots ,{\omega }_k^{\prime })$, there is the unique $g\in G$ such that $({\omega }_1,\cdots ,{\omega }_k{)}^g=({\omega }_1^{\prime },\cdots ,{\omega }_k^{\prime })$.

Here are two examples.

Let $G={\mathrm{PGL}}_n(q)$, and let $V={\mathbb{F}}_q^n$. Then $G$ is sharply $3$-transitive on projective line iff $n=2$.

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• $\mathrm{Aut}(G)=G:⟨\tau ⟩$ where $\tau$ is the Frobenius automorphism acting on ${\mathbb{Z}}_{p^f-1}$.

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