Does the symmetric group $\mathfrak{S}_9$ contain a noncyclic subgroup of order 9?

Question

In general does $\mathfrak{S}_{p^2}$ contain a noncyclic subgroup of order $p^2$?

Answer 1

Just take the cycle $(1 \, 2\, 3 \dots p^2)$, the cyclic group generated will be of order $p^2$.

Answer 2

Consider the subgroup generated by the cycles $(1,2,\ldots,p)$ and $(p+1,p+2,\ldots,2p)$.