en>GabeIglesia: Bolded the abbreviation, per MOS:BOLDTITLE

2014-03-25T14:01:46Z

Bolded the abbreviation, per MOS:BOLDTITLE

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	~~In [[mathematics]]~~, ~~specifically in [[group theory]], '''residue-class-wise affine'''~~		Hi there, I am Alyson Pomerleau and I think it seems fairly good when you say it. Some time in the past she selected to reside in Alaska and her mothers and fathers reside close by. To climb is something she would by no means give up. Office supervising is my profession.<br><br>My web-site ... [http://fashionlinked.com/index.php?do=/profile-13453/info/ psychic chat online]
	~~[[Group (mathematics)\|groups]] are certain [[permutation groups]] [[Group action\|acting]] on~~
	~~<math>\mathbb{Z}</math> (the [[integer]]s), whose elements are [[bijection\|bijective]]~~
	~~residue-class-wise affine [[Map (mathematics)\|mapping]]s.~~

	~~A mapping <math>f: \mathbb{Z} \rightarrow \mathbb{Z}</math> is called '''residue-class-wise affine'''~~
	~~if there is a nonzero integer <math>m</math> such that the restrictions of <math>f</math>~~
	~~to the [[modular arithmetic#Ring of congruence classes\|residue class]]es~~
	~~(mod <math>m</math>) are all [[affine transformation\|affine]]~~. ~~This means that for any~~
	~~residue class <math>r(m) \in \mathbb{Z}/m\mathbb{Z}</math> there are coefficients~~
	~~<math>a_{r(m)}, b_{r(m)}, c_{r(m)} \~~in ~~\mathbb{Z}</math>~~
	~~such that~~ the ~~[[Restriction (mathematics)\|restriction]] of the mapping <math>f</math>~~
	to ~~the [[Set (mathematics)\|set]] <math>r(m) = \{r + km \| k \~~in ~~\mathbb{Z}\}</math> is given by~~

	~~:<math>f\|_{r(m)}: r(m) \rightarrow \mathbb{Z}, \ n \mapsto~~
	~~\frac{a_{r(m)} \cdot n + b_{r(m)}}{c_{r(m)}}</math>.~~

	~~Residue-class-wise affine groups are [[Countable set\|countable]],~~ and ~~they are accessible~~
	~~to [[Computational group theory\|computational investigations]].~~
	~~Many of them act [[multiply transitive]]ly on <math>\mathbb{Z}</math> or on subsets thereof.~~

	~~A particularly basic type of residue-class-wise affine [[permutation]]s are the~~
	~~'''class transpositions''': given [[Disjoint sets\|disjoint]] residue classes <math>r_1(m_1)</math>~~
	and ~~<math>r_2(m_2)</math>, the corresponding '''class transposition''' is the permutation~~
	~~of <math>\mathbb{Z}</math> which interchanges <math>r_1+km_1</math> and~~
	~~<math>r_2+km_2</math> for every <math>k \in \mathbb{Z}</math> and which~~
	~~[[Fixed point (mathematics)\|fixes]] everything else~~. ~~Here it~~ is ~~assumed that~~
	~~<math>0 \leq r_1 < m_1</math> and that <math>0 \leq r_2 < m_2</math>~~.

	~~The set of all class transpositions of <math>\mathbb{Z}</math> [[Generating set of a group\|generates]]~~
	~~a countable [[simple group]] which has the following properties:~~

	* It is ~~not [[Generating_set_of_a_group#Finitely_generated_group\|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 [[subgroup]]s is closed under taking [[Direct product of groups\|direct products]], under taking [[wreath product]]s with finite groups, and under taking restricted wreath products with the infinite [[cyclic group]].
	* It has finitely generated subgroups which do not have [[Presentation of a group\|finite presentations]].
	* It has finitely generated subgroups with [[Undecidable problem\|algorithmically unsolvable]] [[Decision problem\|membership problem]].
	* It has an [[Uncountable set\|uncountable]] series of simple subgroups which is parametrized by the sets of odd [[Prime number\|primes]].

	~~It is straightforward to generalize the notion of a residue-class-wise affine group~~
	~~to groups acting on suitable [[Ring (mathematics)\|rings]] other than~~ <~~math~~>~~\mathbb{Z}~~<~~/math~~>,
	~~though only little work in this direction has been done so far.~~

	~~See also the [[Collatz conjecture]], which is an assertion about a [[Surjective function\|surjective]],~~
	~~but not [[Injective function\|injective]] residue~~-~~class-wise affine mapping.~~

	~~== References and external links ==~~

	*Stefan Kohl. ~~Restklassenweise affine Gruppen~~. ~~Dissertation, Universität Stuttgart, 2005. [http://d-nb~~.~~info/977164071 Archivserver der Deutschen Nationalbibliothek]~~ [http://~~elib.uni-stuttgart~~.de/~~opus/volltexte/2005/2448/ OPUS-Datenbank(Universität Stuttgart)]~~
	*Stefan Kohl. ~~[http:~~/~~/www.gap~~-~~system.org~~/~~Packages~~/~~rcwa.html RCWA] - Residue-Class-Wise Affine Groups. [http://www.gap-system.org 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. [http://dx.doi.org/10.1007/s00209-009-0497-8]

	~~[[Category:Infinite group theory]]~~
	~~[[Category:Number theory]~~]

en>John of Reading: Typo/general fixing, replaced: the the → the using AWB

2013-12-05T20:39:16Z

Typo/general fixing, replaced: the the → the using AWB

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en>Wayne Slam: Reverted edits by 67.207.45.229 (talk) to last revision by ClueBot NG (HG)

2010-11-19T02:57:47Z

Reverted edits by 67.207.45.229 (talk) to last revision by ClueBot NG (HG)

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en>GabeIglesia: Bolded the abbreviation, per MOS:BOLDTITLE

en>John of Reading: Typo/general fixing, replaced: the the → the using AWB

en>Wayne Slam: Reverted edits by 67.207.45.229 (talk) to last revision by ClueBot NG (HG)