In mathematics, specifically in group theory, residue-class-wise affine groups are certain permutation groups acting on
Z {\displaystyle \mathbb {Z} } (the integers), whose elements are bijective residue-class-wise affine mappings. A mapping f : Z → Z {\displaystyle f:\mathbb {Z} \rightarrow \mathbb {Z} } is called residue-class-wise affine if there is a nonzero integer m {\displaystyle m} such that the restrictions of f {\displaystyle f}
to the residue classes (mod m {\displaystyle m} ) are all affine. This means that for any residue class r ( m ) ∈ Z / m Z {\displaystyle r(m)\in \mathbb {Z} /m\mathbb {Z} } there are coefficients
a r ( m ) , b r ( m ) , c r ( m ) ∈ Z {\displaystyle a_{r(m)},b_{r(m)},c_{r(m)}\in \mathbb {Z} }
such that the restriction of the mapping f {\displaystyle f}
to the set r ( m ) = { r + k m ∣ k ∈ Z } {\displaystyle r(m)=\{r+km\mid k\in \mathbb {Z} \}} is given by
f | r ( m ) : r ( m ) → Z , n ↦ a r ( m ) ⋅ n + b r ( m ) c r ( m ) . {\displaystyle 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 Z {\displaystyle \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 ) {\displaystyle r_{1}(m_{1})}
and r 2 ( m 2 ) {\displaystyle r_{2}(m_{2})} , the corresponding class transposition is the permutation of 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 which fixes everything else. Here it is assumed that
0 ≤ r 1 < m 1 {\displaystyle 0\leq r_{1}<m_{1}} and that 0 ≤ r 2 < m 2 {\displaystyle 0\leq r_{2}<m_{2}} . The set of all class transpositions of Z {\displaystyle \mathbb {Z} } generates a countable simple group which has the following properties:
It is not finitely generated. Every finite group, every free product of finite groups and every free group of finite rank embeds into it. The class of its subgroups is closed under taking direct products, under taking wreath products with finite groups, and under taking restricted wreath products with the infinite cyclic group. It has finitely generated subgroups which do not have finite presentations. It has finitely generated subgroups with algorithmically unsolvable membership problem. It has an uncountable series of simple subgroups which is parametrized by the sets of odd primes. It is straightforward to generalize the notion of a residue-class-wise affine group to groups acting on suitable rings other than Z {\displaystyle \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.
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