It is mentioned in the book "Permutation Groups" by Dixon and Mortimer that a 6-transitive permutation group on $n$ elements is $A\_n$ or $S\_n$ and that all the other
$4$ and $5$-transitive permutation groups are the simple Mathieu groups. In particular
they are contained in $A\_n$. 
My question is whether every 3-transitive permutation group on $n$ elements (other than $S\_n$) is also contained in $A\_n$. I would also appreciate references to the results
I quoted from Dixon & Mortimer because none are provided in the book.



Is every nontrivial 3-transitive permutation group contained in $A_n$?
No. The group $\text{PGL}\_2({\bf Z}/p{\bf Z})$ acts transitively
on the $p+1$ points of the projective line ${}\bmod p$ (and the action
is even sharply $3$-transitive), but is not contained in $A\_{p+1}$.
For example, if $c$ is a primitive residue then $({c \; 0 \atop 0 \; 1})$
has cycle structure $1, 1, p-1$ and is thus an odd permutation.


(I see that as I was typing this **Geoff Robinson** gave a counterexample
of the same kind, using the $9$-element field instead of ${\bf Z}/p{\bf Z}$.)


[*Added later*: and if you know about Mathieu groups then it is clear that
$M\_{10}.2$ and $M\_{22}.2$ are further examples; the former is just
${\text P}\Gamma{\text L}\_2({\bf F}\_9)$ (i.e. $\text{PGL}\_2$
extended by the field automorphism), but $M\_{22}.2$ is new.]


