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I came across the following claim in "Adventures in Group Theory" by David Joyner that I couldn't find a proof for.

If $k>5$ and $G$ is a group acting $k$-transitively on a finite set $X$ then $G$ is isomorphic to $S_m$ or to $A_n$, for some $m \ge k$ or some $n \ge k+2$.

If $k \le 5$ what are the counter examples?

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The complete answer is a bit of a long story. I believe you'll find it here.

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    @Derek and Myself, thanks for supplementing my waffle with precise information.2012-01-18