Residue-class-wise affine group explained

In mathematics, specifically in group theory, residue-class-wise affinegroups are certain permutation groups acting on

Z (the integers), whose elements are bijectiveresidue-class-wise affine mappings.

A mapping

f: Z → Z is called residue-class-wise affineif 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 anyresidue class r(m) \in Z /m Z there are coefficientsa _{r(m)} , b _{r(m)} , c _{r(m)} \in Z such that the restriction of the mapping f to the set r(m) = \{r + km \mid k \in Z \} is given byf| _{r(m)} : r(m) → Z , n \mapsto (a _{r(m)} ⋅ n + b _{r(m)})/(c _{r(m)}) .

Residue-class-wise affine groups are countable, and they are accessibleto computational investigations.Many of them act multiply transitively on

Z or on subsets thereof.

A particularly basic type of residue-class-wise affine permutations are theclass transpositions: given disjoint residue classes

r _{1(m} _{1)} and r _{2(m} _{2)} , the corresponding class transposition is the permutationof Z which interchanges r _{1+km} _{1} andr _{2+km} _{2} for every k \in Z and whichfixes everything else. Here it is assumed that0 \leq r _{1} < m _{1} and that 0 \leq r _{2} < m _{2} .

The set of all class transpositions of

Z generatesa 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 groupto groups acting on suitable rings other than

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. https://dx.doi.org/10.1007/s00209-009-0497-8