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Let G≤Sn be 2-transitive other than An and Sn. Is it possible that there exists N⊲ with N\neq G, N transitive and G/N cyclic?

I am interested mostly in the answer when n is large and also when the group G is 3-transitive.

Lior Bary-Soroker
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    G = S_n and N = A_n? – spin Jan 19 '20 at 11:22
  • Other than that, I forgot to mention it. I will fix the question – Lior Bary-Soroker Jan 19 '20 at 12:18
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    I suppose you want G \neq N as well. For all n \geq 2 there is a classification of n-transitive groups, so I guess a starting point would be looking at these lists for examples. – spin Jan 19 '20 at 12:23
  • Thanks, I fixed that too. Sorry for being sloppy. – Lior Bary-Soroker Jan 19 '20 at 13:51
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    PGL_n(F_q) and PSL_n(F_q) should do the trick I think, the determinant on PGL_n(F_q) is a well-defined element of the group $(F_q^) / (F_q^)^n$, which can be non-trivial (and always non-trivial for n=2, q odd).

    For n=2 you will also get 3-transitivity of G, not N though.

     – Lev Soukhanov Jan 19 '20 at 14:56
  • Now that you've added N\neq G, you can remove A_n as an exception.. – verret Jan 19 '20 at 18:06
  • @GeoffRobinson: Good point, but I think you mean non-trivial normal subgroup. – spin Jan 19 '20 at 18:47

2 Answers2

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The most obvious family of examples is AGL(1,q) for q a prime power.

As spin said in the comments, finite 2-transitive groups are classified. They are all almost simple or of affine type (like the example I gave). The almost simple ones are quite explicitly listed, so you would just have to go through the list. You should get plenty more examples. The classification of affine ones is a little less explicit (see Have finite doubly transitive groups been classified?)

verret
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For G = \operatorname{Aut}(M_{22}) and N = M_{22}, with the action of M_{22} on 22 points you have N \triangleleft G < S_{22}. Here both N and G are 3-transitive, and G/N \cong C_2.

spin
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