Residue-class-wise affine group

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

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

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

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

A particularly basic type of residue-class-wise affine permutations are the class transpositions: given disjoint residue classes [math]\displaystyle{ r_1(m_1) }[/math] and [math]\displaystyle{ r_2(m_2) }[/math], the corresponding class transposition is the permutation of [math]\displaystyle{ \mathbb{Z} }[/math] which interchanges [math]\displaystyle{ r_1+km_1 }[/math] and [math]\displaystyle{ r_2+km_2 }[/math] for every [math]\displaystyle{ k \in \mathbb{Z} }[/math] and which fixes everything else. Here it is assumed that [math]\displaystyle{ 0 \leq r_1 \lt m_1 }[/math] and that [math]\displaystyle{ 0 \leq r_2 \lt m_2 }[/math].

The set of all class transpositions of [math]\displaystyle{ \mathbb{Z} }[/math] 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 [math]\displaystyle{ \mathbb{Z} }[/math], 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.

References and external links

• Stefan Kohl. Restklassenweise affine Gruppen. Dissertation, Universität Stuttgart, 2005. Archivserver der Deutschen Nationalbibliothek OPUS-Datenbank(Universität Stuttgart)
• Stefan Kohl. RCWA – Residue-Class-Wise Affine Groups. GAP package. 2005.
• Stefan Kohl. A Simple Group Generated by Involutions Interchanging Residue Classes of the Integers. Math. Z. 264 (2010), no. 4, 927–938. [1]