Residue-class-wise affine group

In mathematics, specifically in group theory, residue-class-wise affine
groups are certain permutation groups acting on
formula_1 (the integers), whose elements are bijective
residue-class-wise affine mappings.
A mapping formula_2 is called residue-class-wise affine
if there is a nonzero integer formula_3 such that the restrictions of formula_4
to the residue classes
(mod formula_3) are all affine. This means that for any
residue class formula_6 there are coefficients
formula_7
such that the restriction of the mapping formula_4
to the set formula_9 is given by
Residue-class-wise affine groups are countable, and they are accessible
to computational investigations.
Many of them act multiply transitively on formula_1 or on subsets thereof.
A particularly basic type of residue-class-wise affine permutations are the
class transpositions: given disjoint residue classes formula_12
and formula_13, the corresponding class transposition is the permutation
of formula_1 which interchanges formula_15 and
formula_16 for every formula_17 and which
fixes everything else. Here it is assumed that
formula_18 and that formula_19.
The set of all class transpositions of formula_1 generates
a countable simple group which has the following properties:
It is straightforward to generalize the notion of a residue-class-wise affine group
to groups acting on suitable rings other than formula_1,
though only little work in this direction has been done so far.
See also the Collatz conjecture, which is an assertion about a surjective,
but not injective residue-class-wise affine mapping.