Something that is true for every element of $\text{Sym}(\Bbb{N})$

Question

I'm trying to prove that:

Every element of $\text{Sym}(\Bbb{N})$ can be written as $f\circ g$ for some $f,g\in \text{Sym}(\Bbb{N})$ with $f^2=g^2$.

But I can't even prove this for $\text{Sym}_n$. Any ideas?

Answer 1

Hints: Write the permutation as a composite of disjoint cycles, some of which could have infinite length. It is then enough to solve the problem for the individual cycles.

Any cycle of length greater than $2$ is a composite of two elements of order $2$. For example, $(1,2,3,4,5) = f \circ g$ with $f=(2,5)(3,4)$, $g=(1,5)(2,4)$. This also applies to cycles of infinite length.

That just leaves cycles of length $2$, where we can take $f$ to be the cycle itself and $g$ the identity.