Residue class

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} 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
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. 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

OPUS-Datenbank(Universität Stuttgart)

  • Stefan Kohl. RCWA - Residue Class-Wise Affine Groups. GAP package. 2005.
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