3
$\begingroup$

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.

$\endgroup$
0

3 Answers 3

7
$\begingroup$

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.]

$\endgroup$
0
6
$\begingroup$

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.

$\endgroup$
1
  • 1
    $\begingroup$ 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 $\{1, \dots, 10\}$. $\endgroup$
    – Stefan Kohl
    Dec 18, 2013 at 23:41
2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.