Definitions

# Residue class-wise affine group

In mathematics, residue class-wise affine groups are certain permutation groups acting on $mathbb\left\{Z\right\}$ (the integers), whose elements are bijective residue class-wise affine mappings.

A mapping $f: mathbb\left\{Z\right\} rightarrow mathbb\left\{Z\right\}$ 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 $r\left(m\right) in mathbb\left\{Z\right\}/mmathbb\left\{Z\right\}$ there are coefficients $a_\left\{r\left(m\right)\right\}, b_\left\{r\left(m\right)\right\}, c_\left\{r\left(m\right)\right\} in mathbb\left\{Z\right\}$ such that the restriction of the mapping $f$ to the set $r\left(m\right) = \left\{r + km | k in mathbb\left\{Z\right\}\right\}$ is given by

$f|_\left\{r\left(m\right)\right\}: r\left(m\right) rightarrow mathbb\left\{Z\right\}, n mapsto$
frac{a_{r(m)} cdot n + b_{r(m)}}{c_{r(m)}}.

Residue class-wise affine groups are countable, and they are accessible to computational investigations. 'Many' of them act multiply transitively on $mathbb\left\{Z\right\}$ or on subsets thereof. Only relatively basic facts about their structure are known so far.

See also the Collatz conjecture, which is an assertion about a surjective, but not injective residue class-wise affine mapping.