# 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.

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.]

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.

• Yes. -- Specifically, ${\rm PGL}(2,9) \cong \langle (3,10,9,8,4,7,6,5), (1,2,4)(5,6,8)(7,9,10) \rangle$ acts 3-transitively on $\{1, \dots, 10\}$. Dec 18, 2013 at 23:41

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.