Transitive group not necessarily have an $n$-cycle

Question

I know that it is true that if $G$ has an $n$-cycle, then $G$ is transitive as a subgroup of $S_n$, but now I'm trying to find an example why the converse is false.

I've been trying examples with $S_3$ and haven't made much progress.

Answer 1

If $p$ is prime, then every transitive subgroup $G\leq S_p$ contains a $p$-cycle. This is because the order of $G$ is divisible by $p$ by the orbit-stabilizer theorem, hence $G$ contains an element of order $p$ by Cauchy's theorem, and this element must be a $p$-cycle.

To see that this is not the case for general $n$, consider the subgroup $$ \{1,(12)(34),(13)(24),(14)(23)\}$$ of $S_4$.

Answer 2

For even $n\gt2$ the alternating group $A_n$ is transitive but contains no $n$-cycle.