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- In mathematics, specifically in group theory, residue-class-wise affinegroups are certain permutation groups acting on (the integers), whose elements are bijectiveresidue-class-wise affine mappings. A mapping is called residue-class-wise affineif there is a nonzero integer such that the restrictions of to the residue classes(mod ) are all affine. This means that for anyresidue class there are coefficientssuch that the restriction of the mapping to the set is given by . Residue-class-wise affine groups are countable, and they are accessibleto computational investigations.Many of them act multiply transitively on or on subsets thereof. A particularly basic type of residue-class-wise affine permutations are theclass transpositions: given disjoint residue classes and , the corresponding class transposition is the permutationof which interchanges and for every and whichfixes everything else. Here it is assumed that and that . The set of all class transpositions of generatesa 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 groupto groups acting on suitable rings other than ,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. (en)
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- 4029 (xsd:nonNegativeInteger)
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- In mathematics, specifically in group theory, residue-class-wise affinegroups are certain permutation groups acting on (the integers), whose elements are bijectiveresidue-class-wise affine mappings. A mapping is called residue-class-wise affineif there is a nonzero integer such that the restrictions of to the residue classes(mod ) are all affine. This means that for anyresidue class there are coefficientssuch that the restriction of the mapping to the set is given by . The set of all class transpositions of generatesa countable simple group which has the following properties: (en)
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- Residue-class-wise affine group (en)
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