Characterize the elements $\beta$ $\in$ $S_n$such that $\beta=\alpha^2$ for some $\alpha$ $\in$ $S_n$

Question

Characterize the elements $\beta$ $\in$ $S_n$ such that $\beta=\alpha^2$ for some $\alpha$ $\in$ $S_n$.

I did the following: suppose that $\beta=\alpha^2$,then $sgn(\beta)=sgn(\alpha^2)=sgn(\alpha)^2=1$, so $\beta$ is an even permutation. Is there a more complete description?

Answer 1

More detailed hints.

1. If $\alpha$ is a cycle of odd length, then $\alpha^2$ is a cycle of the same length.

2. If $\alpha$ is a cycle of length $2k$, then $\alpha^2$ is the product of two cycles of length $k$.

3. Let $\alpha$ be the product of independent cycles. Then and only then is $\alpha$ the square of some permutation when for every integer $k\geq1$ the number of independent cycles of length $2k$ is even (possibly equal to $0$).