Residue-class-wise affine group

From Wikipedia, the free encyclopedia
  (Redirected from Residue class-wise affine groups)
Jump to: navigation, search

In mathematics, specifically in group theory, 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}/m\mathbb{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 \mid 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.

A particularly basic type of residue-class-wise affine permutations are the class transpositions: given disjoint residue classes r_1(m_1) and r_2(m_2), the corresponding class transposition is the permutation of \mathbb{Z} which interchanges r_1+km_1 and r_2+km_2 for every k \in \mathbb{Z} and which fixes everything else. Here it is assumed that 0 \leq r_1 < m_1 and that 0 \leq r_2 < m_2.

The set of all class transpositions of \mathbb{Z} generates a countable simple group which has the following properties:

It is straightforward to generalize the notion of a residue-class-wise affine group to groups acting on suitable rings other than \mathbb{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[edit]