Transitive subgroups of $S_n$

Question

Let $G$ be a transitive subgroup of $S_n$, is true that $G$ contains a transitive subgroup of order $n$?

I know that is true if $n$ is prime, but i am not sure if it´s true for any $n$

Answer 1

Consider $G=A_4$. This group has a subgroup $H$ of index $6$ (for example $\{1, (12)(34)\}$), and hence $G$ acts on coset space $G/H$, giving a homomorphism from $G$ to $S_6$. Notice that the action is transitive and the kernel of the homomorphism is intersection of conjugates of $H$, which is $1$. Thus the homomorphism is injective.

This shows that $A_4$ is a transitive subgroup of $S_6$. Now you can easily answer your question for this specific example.