For a positive integer $n$, let $p_n$ denote the $n$-th prime number. Further let $f: {\rm Sym}(\mathbb{N}) \rightarrow {\rm Sym}(\mathbb{N})$ be the monomorphism which maps a permutation $\sigma$ to the permutation $f(\sigma)$ which maps any prime number $p_n$ to $p_{n^\sigma}$ and which is a homomorphism of the monoid $(\mathbb{N},\cdot)$.

Denoting by $f^{(k)}$ the $k$-fold iteration of $f$, is it true that the trajectory $(f^{(k)}(\sigma))_{k \in \mathbb{N}}$ is cyclic only if $\sigma$ is the identity?

Further, is it even true that $$\bigcap_{k=0}^\infty f^{(k)}({\rm Sym}(\mathbb{N})) \ = \ 1?$$ If the latter is false, can one explicitly describe a non-identity element of this intersection?

imageof $n$ under $\sigma$. $\endgroup$ – Stefan Kohl Feb 18 '17 at 13:50