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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Residue-class-wise affine group ： ウィキペディア英語版 Residue-class-wise affine group In mathematics, specifically in group theory, residue-class-wise affine groups are certain permutation groups acting on $\backslash mathbb$ (the integers), whose elements are bijective residue-class-wise affine mappings. A mapping $f:\; \backslash mathbb\; \backslash rightarrow\; \backslash mathbb$ 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 $r(m)\; \backslash in\; \backslash mathbb/m\backslash mathbb$ there are coefficients $a\_,\; b\_,\; c\_\; \backslash in\; \backslash mathbb$ such that the restriction of the mapping $f$ to the set $r(m)\; =\; \backslash$ is given by :$f|\_:\; r(m)\; \backslash rightarrow\; \backslash mathbb,\; \backslash \; n\; \backslash mapsto$ \frac} 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 $\backslash mathbb$ which interchanges $r\_1+km\_1$ and $r\_2+km\_2$ for every $k\; \backslash in\; \backslash mathbb$ and which fixes everything else. Here it is assumed that and that . The set of all class transpositions of $\backslash mathbb$ 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 $\backslash mathbb$, 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 == *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. () 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Residue-class-wise affine group」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.