Residue-class-wise affine group

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

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


 * $$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 $$\mathbb{Z}$$ 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 $$\mathbb{Z}$$ which interchanges $$r_1+km_1$$ and $$r_2+km_2$$ for every $$k \in \mathbb{Z}$$ and which fixes everything else. Here it is assumed that $$0 \leq r_1 < m_1$$ and that $$0 \leq r_2 < m_2$$.

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

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.