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
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 has an uncountable series of simple subgroups which is parametrized by the sets of odd primes.
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.