Residue-class-wise affine group

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

A mapping ${\displaystyle f:\mathbb {Z} \rightarrow \mathbb {Z} }$ is called residue-class-wise affine if there is a nonzero integer ${\displaystyle m}$ such that the restrictions of ${\displaystyle f}$ to the residue classes (mod ${\displaystyle m}$) are all affine. This means that for any residue class ${\displaystyle r(m)\in \mathbb {Z} /m\mathbb {Z} }$ there are coefficients ${\displaystyle a_{r(m)},b_{r(m)},c_{r(m)}\in \mathbb {Z} }$ such that the restriction of the mapping ${\displaystyle f}$ to the set ${\displaystyle r(m)=\{r+km\mid k\in \mathbb {Z} \}}$ is given by

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

Residue-class-wise affine groups are countable, and they are accessible to computational investigations. Many of them act multiply transitively on ${\displaystyle \mathbb {Z} }$ or on subsets thereof.

A particularly basic type of residue-class-wise affine permutations are the class transpositions: given disjoint residue classes ${\displaystyle r_{1}(m_{1})}$ and ${\displaystyle r_{2}(m_{2})}$, the corresponding class transposition is the permutation of ${\displaystyle \mathbb {Z} }$ which interchanges ${\displaystyle r_{1}+km_{1}}$ and ${\displaystyle r_{2}+km_{2}}$ for every ${\displaystyle k\in \mathbb {Z} }$ and which fixes everything else. Here it is assumed that ${\displaystyle 0\leq r_{1}<m_{1}}$ and that ${\displaystyle 0\leq r_{2}<m_{2}}$.

The set of all class transpositions of ${\displaystyle \mathbb {Z} }$ generates a countable simple group which has the following properties:

• It is not finitely generated.
• Every finite group, every free product of finite groups and every free group of finite rank embeds into it.
• The class of its subgroups is closed under taking direct products, under taking wreath products with finite groups, and under taking restricted wreath products with the infinite cyclic group.
• It has finitely generated subgroups which do not have finite presentations.
• It has finitely generated subgroups with algorithmically unsolvable membership problem.
• 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 ${\displaystyle \mathbb {Z} }$, 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.