Properties of symmetric group $S_n$

Question

Is it true that for each $x \in S_n$, there exist a cycle $y \in S_n$ such that $x = y^k$ for some $k \in \mathbb N$.

I am doing a work on semigroups and their associated graphs. I want to use this result.

Answer 1

No. Take $(12)(345)$ in $S_5$. This element has order $6$ but any cycle has order $5$ or less (and therefore any power of a cycle has order 5 or less).

Answer 2

No. I believe that the simplest example where this is thee case is $S_5$. In particular, the assumption would apply that $S_5 \cong \prod_{i=1}^{n} A_i$ where $A_i$ are cyclic abelian, but $S_5$ is not solvable.