If the Galois group $G$ of a polynomial of degree $n$ is 3-transitive, what can we conclude from this? Is $G=S_n$, where $S_n$ is the symmetric group of order $n$

Any Hint is useful. Thanks

[relocated the following comments by the OP here, JL]

I was reading Topics in Galois Theory by Serre. In theorem 4.4.3., says Let G be a transitive subgroup of $S_n$, which contains transposition. Then TFAE. 1)$ G$ contains $(n-1)$ cycle, 2)$G$ is doubly transitive 3) $G=S_n$.

So was wondering does any such result exist for triple transitive?

  • 2
    $\begingroup$ The alternating group is 3-transitive for most $n$ ($n\ge 5$). $\endgroup$ – Mark Bennet Apr 9 '18 at 12:16
  • $\begingroup$ Questions are usually better-received here if you provide evidence of your own efforts and where you are stuck. $\endgroup$ – samerivertwice Apr 9 '18 at 15:37
  • $\begingroup$ Every finite group can be realized as a Galois group of some extension of fields (conjecturally also as a Galois group $Gal(L/\Bbb{Q})$ but that is still open). Therefore it is a bit unclear what could be said. At least I don't see anything, but I'm not an expert on this. Also, because a Galois group can be seen as a permutation group in many ways (the said field extension is often the splitting field of many polynomials of varying degrees), it is a bit difficult to see how to take advantage. I'm not saying that nothing interesting would not be out there, but... $\endgroup$ – Jyrki Lahtonen Apr 10 '18 at 3:54
  • $\begingroup$ I once had the pleasure of attending a lecture by Abhyankar about realizing the 5-transitive Mathieu groups as Galois groups (in characteristic $p$). He used "derivatives" in some absolutely stunning way, but time has erased all the details from my brain. $\endgroup$ – Jyrki Lahtonen Apr 10 '18 at 3:56
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    $\begingroup$ The finite 3-transitive groups have essentially been classified, using the classification of finite simple groups. See here or here for example. $\endgroup$ – Derek Holt Apr 12 '18 at 12:57

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