Let $S\_n$ be the permutation group on $n$ elements.
Denote by $K(n)$ the largest $k$ s.t. $S\_n$ has a $k$-transitive subgroup (w.r.t. its action on the $n$-element set on which $S\_n$ acts) different from
$S\_n,A\_n$.


I heard that it has been proved that $K(n)\le 7$ for all $n$ but the proof uses
the classification of finite groups (please correct me if I am wrong on this).
But I am also interested in the exact value of $K(n)$ for small $n$ (or even all $n$ if
there is a simple description).


Also I wonder what can be proved in a more elementary way (even weak bounds like $K(n)<n/3$ are interesting).



Maximal $k$-transitivity for a proper subgroup of $S_n$
Actually $K(n)\le5$, with equality only for $n=12$ and $n=24$ and subgroup the Mathieu group $M\_n$. If $G\le S\_n$ is at least $2$-transitive, then an old theorem of Burnside shows that either $G$ has a regular elementary abelian normal subgroup, or $G$ has a simple non-abelian normal subgroup.


In the former case, it is an easy exercise to see that $G$ is at most $3$-transitive,


To treat the latter case indeed requires the classification of the finite simple groups. There isn't even any bound $c$ with $K(n)\le c$ available without the classification.


A nice overview is in [Peter Cameron's article](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.122.1628).


