Residue-class-wise affine group
In mathematics, specifically in group theory, residue-class-wise affine groups are certain permutation groups acting on (the integers), whose elements are bijective residue-class-wise affine mappings.
A mapping is called residue-class-wise affine if there is a nonzero integer such that the restrictions of to the residue classes (mod ) are all affine. This means that for any residue class there are coefficients such that the restriction of the mapping to the set is given by
A particularly basic type of residue-class-wise affine permutations are the class transpositions: given disjoint residue classes and , the corresponding class transposition is the permutation of which interchanges and for every and which fixes everything else. Here it is assumed that and that .
- 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 , though only little work in this direction has been done so far.
- Stefan Kohl. Restklassenweise affine Gruppen. Dissertation, Universität Stuttgart, 2005. Archivserver der Deutschen Nationalbibliothek OPUS-Datenbank(Universität Stuttgart)
- Stefan Kohl. RCWA – Residue-Class-Wise Affine Groups. GAP package. 2005.
- Stefan Kohl. A Simple Group Generated by Involutions Interchanging Residue Classes of the Integers. Math. Z. 264 (2010), no. 4, 927–938.