In mathematics, specifically in group theory, residue-class-wise affine groups are certain permutation groups acting on
(the integers), whose elements are bijective residue-class-wise affine mappings.
is called residue-class-wise affine if there is a nonzero integer
such that the restrictions of
to the residue classes (mod
) are all affine.
This means that for any residue class
there are coefficients
such that the restriction of the mapping
to the set
is given by Residue-class-wise affine groups are countable, and they are accessible to computational investigations.
Many of them act multiply transitively on
or on subsets thereof.
A particularly basic type of residue-class-wise affine permutations are the class transpositions: given disjoint residue classes
, the corresponding class transposition is the permutation of
and which fixes everything else.
The set of all class transpositions of
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
, 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.