# reference on classfication of multiply transitive permutation groups

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?

• @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. – Igor Rivin Nov 4 '15 at 13:33
• 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. – Derek Holt Nov 4 '15 at 13:37
• @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. – Igor Rivin Nov 4 '15 at 13:57
• @DerekHolt And "I guess you have to work out which" speaks exactly to my point. – Igor Rivin Nov 4 '15 at 13:57
• 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 $P{\Gamma}L(2,q)$. – Derek Holt Nov 4 '15 at 15:13

## 1 Answer

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: $A_n$ ($n \ge k+2$), $S_n$ ($n \ge k$).

• 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? – YCor Nov 5 '15 at 10:28
• @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. – Derek Holt Nov 5 '15 at 10:38