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

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?

• Not every element in $A_n$ is a square; you have a necessary condition, but it is not sufficient, so you have not "characterized" the elements yet. Nov 3, 2023 at 17:38
• What happens with a single cycle if you square it? Why can the even permutation $(1,2)(3,4,5,6,7,8)$ not be a square, but $(1,2,3,4)(5,6,7,8)$ is? Nov 3, 2023 at 17:42

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 $$0$$).