Is the torsion set of the group of permutations on $\mathbb N$ closed under composition?

Question

Let $S(\mathbb N)$ be the permutation group over $\mathbb N$ , then is it true that there exist elements $f,g\in S(\mathbb N)$ of finite order such that $f\circ g$ is not of finite order ?

Answer 1

The map $x\mapsto -x$ and the map $x\mapsto 1-x$ of Z to itself have both finite order and their composition does not. On the other hand, there is a bijection between Z and N. These two facts should help you answer your question.