$k$-transitive group actions for $k>5$

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?

• This seems an odd way to thank me for what I believe is a complete answer to your question, what are the examples of $k$-transitive groups for $k\le5$? Perhaps it doesn't give any proofs, but given how long the list of examples is, you are asking for a lot if you want the proofs as well. But are there no references given at the end of the link, and might the proofs you want be found in those references? – Gerry Myerson Jan 18 '12 at 11:51
• If I recall correctly, the fact that if $k>5$, then only $S_k$ and $A_k$ can be $k$-transitive is quite doable. Also the fact that if $k=4$ or $k=5$ only the Mathieu-groups join the picture is not so hard. But of course, the classification of 2-transitive groups is hard. – Myself Jan 18 '12 at 14:56
• There are no proofs known that do not involve the classification of finite simple groups. It is not particularly hard to show (the O'Nan-Scott Theorem does this) that an unknown $k$-transitive group for $k \ge 3$ say (or $k \ge 2$ with a bit more work) would be almost simple (= extension of nonabelian simple group by automorphisms) so, given CFSG, it just reduces to checking that the known almost simple groups have no such representations, which was done long before CFSG was proved. – Derek Holt Jan 18 '12 at 21:10