Residue class-wise affine group

In mathematics, 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 $f:\; mathbb\{Z\}\; ightarrow\; mathbb\{Z\}$ 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 any residue class $r(m)\; inmathbb\{Z\}/mmathbb\{Z\}$ 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 $r(m)\; =\; \{r\; +\; km$
k in mathbb{Z}} is given by

:$f|\_\{r(m)\}:\; r(m)\; ightarrow\; mathbb\{Z\},\; n\; mapstofrac\{a\_\{r(m)\}\; cdot\; n\; +\; b\_\{r(m)\{c\_\{r(m)$.

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.Only relatively basic facts about their structure are known 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. [http://deposit.ddb.de/cgi-bin/dokserv?idn=977164071 Archivserver Deutsche Bibliothek] [http://elib.uni-stuttgart.de/opus/volltexte/2005/2448/ OPUS-Datenbank(Universität Stuttgart)]
*Stefan Kohl. [http://www.gap-system.org/Packages/rcwa.html RCWA] - Residue Class-Wise Affine Groups. [http://www.gap-system.org GAP] package. 2005.