Residue-class-wise Affine Group

In mathematics, specifically in group theory, residue-class-wise affine
groups are certain permutation groups acting on
$\mathbb$ (the integers), whose elements are bijective
residue-class-wise affine mappings.

A mapping $f: \mathbb \rightarrow \mathbb$ is called residue-class-wise affine
if there is a nonzero integer $m$ such that the restrictions of $f$
to the residue classes
(mod $m$) are all affine. This means that for any
residue class $r(m) \in \mathbb/m\mathbb$ there are coefficients
$a_, b_, c_ \in \mathbb$
such that the restriction of the mapping $f$
to the set $r(m) = \$ is given by

:$f, _: r(m) \rightarrow \mathbb, \ n \mapsto \frac$.

Residue-class-wise affine groups are countable, and they are accessible
to computational investigations.
Many of them act multiply transitively on $\mathbb$ or on subsets thereof.

A particularly basic type of residue-class-wise affine permutations are the
class transpositions: given disjoint