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?