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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.## References and external links

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 $r(m)\; in\; mathbb\{Z\}/mmathbb\{Z\}$ 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

- $f|\_\{r(m)\}:\; r(m)\; rightarrow\; mathbb\{Z\},\; n\; mapsto$

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.

- Stefan Kohl. Restklassenweise affine Gruppen. Dissertation, Universität Stuttgart, 2005. Archivserver Deutsche Bibliothek

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Last updated on Wednesday May 30, 2007 at 13:26:32 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Wednesday May 30, 2007 at 13:26:32 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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