Residue-class-wise affine group

{{Short description|Group theory}}

In mathematics, specifically in group theory, residue-class-wise affine

groups are certain permutation groups acting on

$\backslash mathbb\{Z\}$ (the integers), whose elements are bijective

residue-class-wise affine mappings.

A mapping $f:\; \backslash mathbb\{Z\}\; \backslash rightarrow\; \backslash mathbb\{Z\}$ 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 $r(m)\; \backslash in\; \backslash mathbb\{Z\}/m\backslash mathbb\{Z\}$ there are coefficients

$a\_\{r(m)\},\; b\_\{r(m)\},\; c\_\{r(m)\}\; \backslash in\; \backslash mathbb\{Z\}$

such that the restriction of the mapping $f$

to the set $r(m)\; =\; \backslash \{r\; +\; km\; \backslash mid\; k\; \backslash in\; \backslash mathbb\{Z\}\backslash \}$ is given by

:$f|\_\{r(m)\}:\; r(m)\; \backslash rightarrow\; \backslash mathbb\{Z\},\; \backslash \; n\; \backslash mapsto$

\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 $\backslash mathbb\{Z\}$ or on subsets thereof.

A particularly basic type of residue-class-wise affine permutations are the

class transpositions: given disjoint residue classes $r\_1(m\_1)$

and $r\_2(m\_2)$, the corresponding class transposition is the permutation

of $\backslash mathbb\{Z\}$ which interchanges $r\_1+km\_1$ and

$r\_2+km\_2$ for every $k\; \backslash in\; \backslash mathbb\{Z\}$ and which

fixes everything else. Here it is assumed that

and that .

The set of all class transpositions of $\backslash mathbb\{Z\}$ 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 $\backslash mathbb\{Z\}$,

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.