Is it possible to write every even permutation in $S_n$ as a square of some even permutation?

Question

In other words, consider $A_n$, the alternating group of the $n$-th symmetrical group $S_n$, is it true that $$A_n=\{a^2\mid a\in A_n\}$$? I tested for $S_3$ and it seemed to hold. If it is true, it will be very helpful to me for solving another problem.

Answer 1

darij is incorrect; it's not even possible to write every even permutation as a square of some permutation, even or odd! As an exercise, show that a permutation has a square root (in $S_n$) iff it has an even number of cycles of each even length. (Derek Holt's example in the comments is the smallest example where a permutation has cycles of two different even lengths.)

Answer 2

You are asking whether the map $A_n \rightarrow A_n$ defined by $a \mapsto a^2$ is surjective. Since $A_n$ is a finite set that's the same asking whether the map is bijective. This is not true when $n \geq 4$, because then the map is not injective: for example, you can find $a \neq 1$ with $a^2 = 1$.