Is every nontrivial 3-transitive permutation group contained in $A_n$? 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.
 A: 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.]
A: Isn't the group ${\rm PGL}(2,9)$ triply transitive on the 10 $1$-dimensional subspaces of a $2$-dimensional vector space over ${\rm GF}(9)$, and it contains an $8$-cycle, so contains an odd permutation.
A: Some references:
A.I.Sozutov, On groups with almost perfect involution, Proceedings of the Steklov Institute of Mathematics , Volume 257(2007), pp. S181-S188
Kerby W., Wefelscheid H. Uber eine scharf 3-fach transitiven Gruppen zugeordnete algebraische Struktur. Abh. Math. Sem. Univ. Hamburg 37 (1972), pp. 225-235.
