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A permutation group $G$ on a set $X$ is said to be $k$-transitive if it is both transitive on $X$ and either $k=1$ or the point stabilizer $G_x$ is $(k-1)$-transitive on $X\setminus\{x\}$.

Is there a classification of 3-transitive finite groups?

Examples:

  • For every non-negative integer $n$, $S_n$ and $A_n$ are 3-transitive on $\{1,2,\dots,n\}$
  • If $n-1=p^f$ is a prime power, then all $G$ with $\operatorname{PGL}(2,p^f) \leq G \leq \operatorname{P\Gamma L}(2,p^f)$ are 3-transitive
  • If $n-1=q^2$ is the square of a prime power, then also all $G$ with $M(q^2)\leq G \leq \operatorname{P\Gamma L}(2,p^f)$, where $\operatorname{PSL}(2,q^2) < M(q^2) < \operatorname{P\Gamma L}(2,q^2)$

Every sharply triply transitive group is either $\operatorname{PGL}(2,p^f)$ or $M(q^2)$, of order $((n-1)^2-1)((n-1)^2-(n-1))/(n-2) = n(n-1)(n-2)$. This is due to Zassenhaus; see Huppert–Blackburn (XI.1.4.b, XI.2.1, and XI.2.6). However, there are triply transitive groups that are not sharply triply transitive (such as $\operatorname{P\Gamma L}(2,p^f)$ for $f>1$).

If $n$ is odd, then Wagner (1966) showed that any non-identity normal subgroup of a triply transitive group is also triply transitive. By taking a minimal normal subgroup (and then a minimal normal subgroup of that) we get a simple triply transitive group of the same degree, so if we are only interested in $n$, then we need only consult our knowledge of finite simple groups.

I think $\operatorname{ASL}(n,2)$ is always triply transitive.

Here are the triply transitive groups of degree $n < 2500$ that don't fall into the above categories:

  • $M_{11}$ of degree 11
  • $M_{11}$ of degree 12, $M_{12}$ of degree 12
  • $A_7 \ltimes 2^4$ of degree 16 (?)
  • $M_{22}, \operatorname{Aut}(M_{22})$ of degree 22
  • $M_{23}$ of degree 23
  • $M_{24}$ of degree 24

Note these are mostly Mathieu groups.

Bibliography

  • Wagner, A. “Normal subgroups of triply-transitive permutation groups of odd degree.” Math. Z. 94 (1966) 219–222 MR199251 DOI:10.1007/BF01111350
  • Siemons, Johannes. “Normal subgroups of triply transitive permutation groups of degree divisible by 3.” Math. Z. 174 (1980), no. 2, 95–103. MR592907 DOI:10.1007/BF01293530
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  • $\begingroup$ XII.10.5 (Klemm, 1977) ams.org/mathscinet-getitem?mr=444751 may be relevant $\endgroup$ – Jack Schmidt Mar 4 '14 at 1:03
  • $\begingroup$ I wrote up the 2-transitive groups when studying strongly p-embedded subgroups: math.stackexchange.com/questions/16954/… $\endgroup$ – Jack Schmidt Mar 4 '14 at 4:50
  • $\begingroup$ one simple observation.Assume that you found a subgroup $H$ of $G$ which is transitive on $X$. Then it is easy to show that $G=H.G_x$ in that case $G$is 3-transitive,$H$ is transive and $G_x$ two transitive. $\endgroup$ – mesel Mar 4 '14 at 8:13
  • $\begingroup$ What I really mean , Start with a two transitive group then add a one elemen to acted set,Find an another set which is transitive on new set, If you define a semiderect produc of these two group such that two transitive group behave like a stabilizer,then you will get three transitive group. I hope the idea is clear. $\endgroup$ – mesel Mar 4 '14 at 8:20
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    $\begingroup$ I don't see any omissions from your list. It shouldn't be too hard to prove formally that all triply transitive subgroups of ${\rm P \Gamma L}(2,q)$ contain ${\rm PGL}(2,q)$ or $M(q)$. ${\rm ASL}(n,2)= {\rm AGL}(n,2)$ is triply transitive because ${\rm SL}(n,2)$ is $2$-transitive on vectors, but you can see from the list of $2$-transitive groups that it has no proper $2$-transitive subgroups except for $A_7 < {\rm SL}(4,2)$. There are no other affine examples. $\endgroup$ – Derek Holt Mar 4 '14 at 8:52

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