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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?

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  • $\begingroup$ What is $n^\sigma$? Should it be $\sigma_n$ or $\sigma(n)$? $\endgroup$ – Max Alekseyev Feb 18 '17 at 13:47
  • $\begingroup$ @MaxAlekseyev $n^\sigma$ denotes the image of $n$ under $\sigma$. $\endgroup$ – Stefan Kohl Feb 18 '17 at 13:50
  • $\begingroup$ Ok, so it's same as $\sigma(n)$. $\endgroup$ – Max Alekseyev Feb 18 '17 at 14:28
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If $\sigma$ is the identity, then so is $f(\sigma)$.

Suppose that $\sigma$ is not the identity. Let $m_k$ be the smallest non-fixed point of $f^{(k)}(\sigma)$. It is clear that $m_0\geq 1$ is some finite integer, and for any $k\geq 0$, $m_{k+1} = p_{m_k} > m_k$. In particular, we have $m_k\geq q_k$, where $q_0=1$ and $q_{i+1}=p_{q_i}$ for $i\geq 0$ (A007097).

It follows that the trajectory of $f^{(k)}(\sigma)$ cannot be cyclic unless $\sigma$ is the identity.

Now, for every $k\geq 0$, all elements of $f^{(k)}(\mathrm{Sym}(\mathbb N))$, except the identity, have smallest non-fixed point $\geq q_k$. Hence, the intersection of $f^{(k)}(\mathrm{Sym}(\mathbb N))$ consists of the identity permutation only.

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  • $\begingroup$ @StefanKohl: I'm not sure what is interesting to you. Personally I find it quite interesting that $m_k$ is fully determined by $m_0$ (i.e., the smallest non-fixed point of $\sigma$) and otherwise does not depend on $\sigma$. $\endgroup$ – Max Alekseyev Feb 18 '17 at 19:53

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