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Suppose $G$ is a finite group, and suppose that it acts $2$-transitively in each of the permutation representations $(G,X_i)$ ($i$ ranges over some index set $I$), where the $X_i$s all have different size.

  1. First question: what can be said about $G$ and/or $I$ without the Classification of Finite Simple Groups (CFSG) ?
  2. Second question: what can be said about $G$ and/or $I$ with the use of CFSG ?
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By Burnside, a finite doubly transitive group is either a subgroup of $\text{AGL}_n(p)$, or a subgroup of $\text{Aut(S)}$ for a finite non-abelian simple group $S$. In the former case, one quickly checks that there is no faithful doubly transitive action of degree $\ne p^n$.

Without CFSG, I believe not much can be said in the second case. There are a few cases with $|I|=2$, like for the Mathieu group $\text{M}_{11}$ which acts doubly transitively on $11$ and $12$ points. And a few more examples coming from exceptional isomorphisms of some small simple groups, like $\text{PSL}_2(7)\cong\text{PSL}_3(2)$, acting naturally and doubly transitively on $8$ and $7$ points, respectively.

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