Is a transitive subgroup $H \leq S_n$ of cardinality $n$ automatically cyclic?

Question

It is very reasonable for me to assume that if $H \leq S_n$ is transitive and $|H|=n$, then $H$ must be generated by a $n$-cycle, but I cannot seem to be able to prove it, so I really can not be sure that it is true. Could you please help me with that? Thank you!

EDIT: Thank you anyone! Could you tell me if we can at least say anything about the group being abelian? How about solvable?

Answer 1

The smallest counterexample is $\{1,(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)\}\leq S_4$.

Indeed, any group of order $n$ is a transitive subgroup of $S_n$ by Cayley's theorem.

Answer 2

This is not true. Consider the following subgroup of $S_4$:

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

It is transitive and of order $4$ but it isn't cyclic.