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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?

Igor Rivin
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  • j.p.'s answer in http://mathoverflow.net/q/32351 mentions the book by Dixon and Mortimer – Geoff Robinson Nov 04 '15 at 13:25
  • @GeoffRobinson I am looking at D&M as we speak, and can't seem to find a clean statement. – Igor Rivin Nov 04 '15 at 13:29
  • Yes, they are listed in Section 7.7 of Dixon & Mortimer's book. I remember seeing this list as a research student in about 1971. That was before CFSG of course, but no new 2-transitive groups have been found since then. Also, I think there ahs been a more complete description of the 2-transitive affine groups since then. – Derek Holt Nov 04 '15 at 13:30
  • ??? Section 7.7 is called "The Finite 2-transitive Groups". What more could you want? Well, I suppiose you need to chase a couple of references to get a precise description of the affine groups. – Derek Holt Nov 04 '15 at 13:32
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    @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 04 '15 at 13:33
  • There seems to be a good list of transitive linear groups on Wikipedia, which is relevant to the affine case. – Geoff Robinson Nov 04 '15 at 13:33
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    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 04 '15 at 13:37
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    Well, the $2$-transitive list should include all more highly transitive cases. It is really then a question of deciding which $2$-transitive groups are three transitive, since there are only a handful of non-symmetric/alternating groups more than 3-transitive. – Geoff Robinson Nov 04 '15 at 13:38
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    @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 04 '15 at 13:57
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    @DerekHolt And "I guess you have to work out which" speaks exactly to my point. – Igor Rivin Nov 04 '15 at 13:57
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    http://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 04 '15 at 15:13
  • The book on permutation groups by Peter Cameron should also have things somewhat sorted out. – Benjamin Steinberg Nov 05 '15 at 01:09
  • @BenjaminSteinberg, I was going to recommend Cameron's book also -- the list is entirely explicit (as opposed to "discursive"). However I had a very vague memory that there was an error somewhere in one of those tables -- I could be completely wrong, but perhaps it would be worth comparing Cameron's tables with those in D&M to make sure. – Nick Gill Nov 05 '15 at 10:02
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    I just had a quick look at the two lists, and I think D&M might be missing the $2$-transitive group $P{\Gamma}L(2,8)$ in degree $28$. But Cameron's list has the same problem (which Igor complained about) as D&M's that he doesn't go into details about exactly which subgroups of $P{\Gamma}L(2,q)$ are $3$-transitive. – Derek Holt Nov 05 '15 at 10:26
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    There are actually $4$ sharply $2$-transitive groups of degree $121$. After staring at Cameron's list for a while, I think they are all there, but his descriptions are not completely precise, and he doesn't specify the precise number of groups. – Derek Holt Nov 05 '15 at 12:16
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    @GeoffRobinson To be more precise, he has classes labelled ${\rm SL}(2,5) \unlhd H$ (where $H$ is the stabilizer), ${\rm SL}(2,3) \unlhd H$, and a large class of groups of degree $q^d$ labelled ${\rm SL}(d,q) \le H \le {\Gamma}L(d,q)$. I think there are two sharply $2$-transitive groups of degree $121$ that come under ${\rm SL}(1,q) \le H \le {\Gamma}L(1,q)$. – Derek Holt Nov 05 '15 at 12:44

2 Answers2

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

Derek Holt
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  • 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 05 '15 at 10:28
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    @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 05 '15 at 10:38
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This is a bit absurd as a reference, but if you just want an explicit list that has the dubious merit of having gotten past a referee, the $3$-transitive groups are listed on pp. 86-87 of Abhyankar's paper Galois theory on the line in nonzero characteristic. That list also explicitly states which groups in the list are $4$-transitive, $5$-transitive, etc. On the other hand, a drawback of Abhyankar's list is that it's missing some of the groups between $\text{PSL}_2(q)$ and $\text{P}\Gamma\text{L}_2(q)$ when $q$ is an odd fourth power. Personally I prefer the list in Derek Holt's answer, since I value correctness more than publishedness.

Michael Zieve
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