Transitive action of normal subgroup of the alternating group

Question

everyone! Would anyone be willing to give me any sort of help with the following question?

Let $n\ge 4$ and $A_n$ the alternating group. Let $N$ a non-trivial normal subgroup of $A_n$. Prove that the action of $N$ on $\{1,2,...,n\}$ is transitive.

Let me stress that one is NOT allowed to use the fact that $A_n, n\ge 5$ is simple.

Any help will be greatly appreciated!

MORE INFORMATION: Would showing that if $X$ is an $N-orbit$ then $gX$ is an $N$-orbit, where $g \in A_n$ and using Cardinality of a subset acted upon by the Alternating Group, $A_n$ help?

Answer 1

Take a nontrivial element $g\in N$. Suppose $g$ sends 1 to 2. For any $x\in\{3,\ldots,n\}$, note that $(2\ x)g(2\ x)^{-1}$ is an element of $N$ that sends 1 to $x$. So the orbit of 1 is $\{1,\ldots,n\}$, hence $N$ acts transitively.

Edit: Pardon my stupidity. Assuming $x\ne 3,4$, you could consider $((2\ x)(3\ 4))g((2\ x)(3\ 4))^{-1}$ instead. That'll leave out the case $n=4$ though.

Answer 2

Well, since $\,A_n\,\,\,\,\,n\geq 5\,$ , is simple, the only example there is for your question is $\,A_4\,$ , and the only normal non trivial subgroup of this one is

$$\{(1)\,,\,(12)(34)\,,\,(13)(24)\,,\,(14)(23)\}$$

which is easily seen to be transitive on $\,\{1,2,3,4\}\,$