Residue-class-wise affine group

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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.

A mapping 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 and , the corresponding class transposition is the permutation of which interchanges and for every and which fixes everything else. Here it is assumed that and that .

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.

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