5
$\begingroup$

It seems to be true that multiply transitive permutation groups have been classified completely (using CFSG), but I am having trouble finding a reference where this classification is actually stated. Is there a canonical reference?

$\endgroup$
  • 2
    $\begingroup$ @DerekHolt If by "multiply transitive" you mean "2-transitive", then you don't want anything else. I see no list of 3-transitive groups there. $\endgroup$ – Igor Rivin Nov 4 '15 at 13:33
  • 2
    $\begingroup$ The descriptions tell you which are 3-transitive! These are $A_n,S_n$, some affine groups of eevn degree, some subgroups of $P{\Gamma}L(2,q)$ (I guess you have to work out which), the Mathieu groups. $\endgroup$ – Derek Holt Nov 4 '15 at 13:37
  • 3
    $\begingroup$ @GeoffRobinson I agree. The thing is that even the list in D&M is very discoursive. If I just want to put in a reference in a paper saying: these are the 2- 3- 4- 5- transitive groups (see \cite{greatReference}), someone who actually cares would have to spent a fair bit of time trying to sort things out in D&M. It would be nice just to have a Landau-style telegraphic statement. $\endgroup$ – Igor Rivin Nov 4 '15 at 13:57
  • 5
    $\begingroup$ @DerekHolt And "I guess you have to work out which" speaks exactly to my point. $\endgroup$ – Igor Rivin Nov 4 '15 at 13:57
  • 3
    $\begingroup$ math.stackexchange.com/questions/698327 seems to give an accurate description of the finite $3$-transitive groups, although $A_n$ and $S_n$ are $3$-transitive only when $n \ge 5$ and $n \ge 3$, respectively. I still don't think you can improve on D&M for the 2-transitive groups, and 4- and 5-transitive lists are easy. I admit that 3-transitive is a little trickier to get right, mainly because of the complications with $P{\Gamma}L(2,q)$. $\endgroup$ – Derek Holt Nov 4 '15 at 15:13
12
$\begingroup$

Here is a list of the finite $3$-transitive groups, derived by looking through the list of $2$-transitive groups in Section 7.7 of Dixon and Mortimer and identifying those that are $3$-transitive.

Let's first recall the structure of $G := P{\Gamma}L(2,q)$ with $q=p^e$, $p$ prime. Let $S={\rm PSL}(2,q)$. For $q$ even, $G = S \rtimes \langle \phi \rangle$ with $\phi$ acting as field automorphism of order $e$, and $G/S \cong C_e$.

For $q$ odd, $G = S\langle \delta,\phi \rangle$, where $\delta$ acts as a diagonal automorphism of order $2$, and $\phi$ as a field automorphism, (Note that this extension is nonsplit when $e$ is even.) We have $G/S \cong C_2 \times C_e$. The subgroup $ S \langle \phi \rangle$ of index $2$ in $G$ is denoted by $P{\Sigma}L(2,q)$.

So now, the finite $3$-transitive groups are as follows.

$A_n$ ($n \ge 5$), $S_n$ ($n \ge 3$), degree $n$. (There are two inequivalent actions, conjugate under an outer automorphism of $S_6$, when $n=6$.)

$A{\Gamma}L(n,2) = {\mathbb F}_2^n \rtimes {\rm GL}(n,2)$ with $n \ge 2$. degree $2^n$.

${\mathbb F}_2^4 \rtimes A_7$, degree $16$.

Groups $G$ with ${\rm PSL}(2,2^e) \le G \le P{\Gamma}L(2,2^e)$, degree $2^n+1$.

For $q$ odd, groups $G$ with ${\rm PSL}(2,q) \le G \le P{\Gamma}L(2,q)$ and $G \not\le P{\Sigma}L(2,q)$, degree $q+1$.

The Mathieu groups $M_{11},M_{12},M_{22},M_{22}.2 = {\rm Aut}(M_{22}), M_{23},M_{24}$, degrees $11,12,22,22,23,24$.

$M_{11}$, degree $12$.

For completeness, the finite $4$-transitive groups are: $A_n$ ($n \ge 6$), $S_n$ ($n \ge 4$), $M_{11},M_{12},M_{23},M_{24}$,.

The $5$-transitive groups are: $A_n$ ($n \ge 7$), $S_n$ ($n \ge 5$), $M_{12},M_{24}$.

And the finite $k$-transitive groups for $k \ge 6$ are: $A_n$ ($n \ge k+2$), $S_n$ ($n \ge k$).

$\endgroup$
  • $\begingroup$ The precise list should mention the pair (group, set with a 3-transitive action) and not only the group. Is it true that in all cases, the stabilizer is unique up to conjugation? if not, up to automorphism? $\endgroup$ – YCor Nov 5 '15 at 10:28
  • 2
    $\begingroup$ @YCor There is a unique conjugacy class of stabilizers in all cases except for $A_6$ and $S_6$ in degree $6$, in which case there are two such classes fused under an auromorphism. I have added a note about that. $\endgroup$ – Derek Holt Nov 5 '15 at 10:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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