Transitive subgroup of $\mathcal{S}_n$ implies normality for that subgroup?

Question

While solving an specific question about the Galois pairing for a quartic polynomial i proved that it's Galois group has order 6 and it couldn't be $\mathcal{A}_4$ and as it acts transitively i jumped to the conclusion that it had to be $\mathcal{S}_4$ which is cleary a sloppy argument.

So my question is: Given $X$ a transitive subgroup of $\mathcal{S}_n$ is $X$ a normal subgroup?

Thanks in advance

Answer 1

This is not true in general. Take $\sigma=(1 \ 2 \dots \ n)$ and $X=\langle \sigma \rangle$ is transitive. And for $n \ge 5$ the only propernormal subgroup of $\mathcal S_n$ is $\mathcal A_n \neq \langle \sigma \rangle$