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Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$. Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition $\tau_{r_1(m_1),r_2(m_2)}$ be the permutation of $\mathbb{Z}$ which interchanges $r_1+km_1$ and $r_2+km_2$ for every $k \in \mathbb{Z}$ and which fixes everything else.

The group ${\rm CT}(\mathbb{Z})$ generated by all class transpositions of $\mathbb{Z}$ is simple (cf. Math. Z. 264 (2010), no. 4, 927-938, http://dx.doi.org/10.1007/s00209-009-0497-8).

Now call a permutation of $\mathbb{Z}$ residue-class-wise affine if there is a positive integer $m$ such that its restrictions to the residue classes (mod $m$) are all affine.

So far there is no way known to decide whether a given residue-class-wise affine permutation is an element of ${\rm CT}(\mathbb{Z})$ or not -- although there is a heuristic factorization method implemented in the GAP package RCWA (cf. http://www.gap-system.org/Packages/rcwa.html) which works in many - also highly nontrivial - cases (cf. Section 5 in the aforementioned paper, where a permutation considered by Collatz in the 1930's is shown to lie in ${\rm CT}(\mathbb{Z})$), but not always!.

It seems not unreasonable to conjecture that ${\rm CT}(\mathbb{Z})$ is the group of all residue-class-wise affine permutations of $\mathbb{Z}$ which fix the nonnegative integers setwise - but is this conjecture true?

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I would certainly call $CT(\mathbf{Z})$ the group of all residue-class-wise affine permutations of $\mathbf{Z}$, while the group you consider (or the possibly larger stabilizer of $\mathbf{N}$) looks like an interesting subgroup, but as a kind of stabilizer. Anyway, let $CT'(\mathbf{Z})$ be the big group: then the action of this large group extends to a continuous action on the profinite completion $\hat{\mathbf{Z}}$. This should help understanding it. –  YCor Nov 15 '12 at 19:20
    
I don't see how this should help -- but maybe you have some good idea? One also has a continuous action of ${\rm CT}(\mathbb{Z})$ on $\mathbb{Z}$, endowed with a topology by taking the set of all residue classes as a basis -- but I don't see so far that this helps further. But maybe you have better luck with your approach. By the way, I prefer the notation ${\rm RCWA}(\mathbb{Z})$ for the group of all residue-class-wise affine permutations (that's the notation I used in my publications so far). –  Stefan Kohl Nov 15 '12 at 21:18
    
By the way -- I asked this question already in 2010 as Problem 17.59 in the Kourovka Notebook. –  Stefan Kohl Nov 22 '12 at 16:55

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