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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?

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    $\begingroup$ 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. $\endgroup$ Nov 3, 2023 at 17:38
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    $\begingroup$ 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? $\endgroup$
    – ahulpke
    Nov 3, 2023 at 17:42

1 Answer 1

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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 $0$).

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