Jordan’s theorem (multiply transitive groups)

Let $G$ be a sharply $n$-transitive permutation group, with $n\ge 4$. Then

1. $G$ is similar to $S_n$ with the standard action or

2. $G$ is similar to $A_{n+2}$ with the standard action or

3. $n=4$ and $G$ is similar to $M_{11}$, the Mathieu group of degree $10$ or

4. $n=5$ and $G$ is similar to $M_{12}$, the Mathieu group of degree $11$.