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In mathematics, specifically in group theory, residue-class-wise affinegroups are certain permutation groups acting on Z {\displaystyle \mathbb {Z} } , whose elements are bijectiveresidue-class-wise affine mappings.
A mapping f : Z → Z {\displaystyle f:\mathbb {Z} \rightarrow \mathbb {Z} } is called residue-class-wise affineif there is a nonzero integer m {\displaystyle m} such that the restrictions of f {\displaystyle f} to the residue classes are all affine. This means that for anyresidue class r ∈ Z / m Z {\displaystyle r\in \mathbb {Z} /m\mathbb {Z} } there are coefficients a r , b r , c r ∈ Z {\displaystyle a_{r},b_{r},c_{r}\in \mathbb {Z} } such that the restriction of the mapping f {\displaystyle f} to the set r = { r + k m ∣ k ∈ Z } {\displaystyle r=\{r+km\mid k\in \mathbb {Z} \}} is given by
Residue-class-wise affine groups are countable, and they are accessibleto computational investigations.Many of them act multiply transitively on Z {\displaystyle \mathbb {Z} } or on subsets thereof.
A particularly basic type of residue-class-wise affine permutations are theclass transpositions: given disjoint residue classes r 1 {\displaystyle r_{1}} and r 2 {\displaystyle r_{2}} , the corresponding class transposition is the permutationof Z {\displaystyle \mathbb {Z} } which interchanges r 1 + k m 1 {\displaystyle r_{1}+km_{1}} and r 2 + k m 2 {\displaystyle r_{2}+km_{2}} for every k ∈ Z {\displaystyle k\in \mathbb {Z} } and whichfixes everything else. Here it is assumed that 0 ≤ r 1 < m 1 {\displaystyle 0\leq r_{1}