Even permutation in Sn How to show that if $\pi \in S_n$ is a square then $\pi$ is an even permutation. Is the converse statement true: each even permutation is a square?

All squares are even because the product of an even permutation with an even permutation is even, and so is the product of an odd permutation with an odd permutation.
On the other hand, not all even permuations are squares. For example $(1\;2)(3\;4\;5\;6)$ is not a square.
A permutation is square if and only if, for every even $k$, its cycle decomposition contains an even number (possibly 0) of cycles of length $k$. (The number of cycles of odd lengths can be arbitrary).
