Residue class-wise affine group

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.


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