# Is there a 2 transitive finite group with transitive normal subgroup having a cyclic quotient other than $A_n$ and $S_n$?

Let $$G \leq S_n$$ be $$2$$-transitive other than $$A_n$$ and $$S_n$$. Is it possible that there exists $$N\lhd G$$ 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.

• $G = S_n$ 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
• 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
• $PGL_n(F_q)$ 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

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 $$G/N \cong C_2$$.
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?)