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Classification of triply transitive finite groups

Mathematics Asked by Jack Schmidt on December 1, 2021

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 $Xsetminus{x}$.

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 $operatorname{PGL}(2,p^f) leq G leq operatorname{PGamma L}(2,p^f)$ are 3-transitive
  • If $n-1=q^2$ is the square of a prime power, then also all $G$ with $M(q^2)leq G leq operatorname{PGamma L}(2,p^f)$, where $operatorname{PSL}(2,q^2) < M(q^2) < operatorname{PGamma L}(2,q^2)$

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 $operatorname{PGamma L}(2,p^f)$ 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

One Answer

In Graham's & All Handbook of Combinatorics, Peter Cameron wrote (however without a proof)

If t=3, then either G is one of more transitive, or

  • G is $ AGL(d,2), n=2^d $ or
  • G is $ V_{16}.A_{7}$ or
  • G is $ M_{11} $ or
  • G is $ M_{22} $ or $Aut(M_{22}) $ or
  • $ PSL(2,q) <= G <= PGamma L(2,q) $, n=q+1, q= prime power

Answered by Slepecky Mamut on December 1, 2021

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