Projective connection - Revision history

en>Frietjes at 23:19, 29 September 2014

2014-09-29T23:19:02Z

← Older revision		Revision as of 01:19, 30 September 2014
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	~~In the [[mathematics\|mathematical]] field of [[group theory]], a [[Group (mathematics)\|group]] ''G'' is '''residually finite''' or '''finitely approximable''' if~~ for ~~every nontrivial element ''g'' in ''G'' there~~ is ~~a [[Group homomorphism\|homomorphism]] ''h'' from ''G'~~' to a ~~finite group, such that~~		They contact me Emilia. The preferred pastime for my kids and me is to play baseball and I'm trying to make it a profession. Hiring has been my profession for some time but I've already applied for another one. For a whilst she's been in South Dakota.<br><br>Also visit my blog post: [http://www.fuguporn.com/blog/36031 at home std testing]

	~~:<math>h(g) \neq 1~~.~~\,</math>~~

	~~There are a number of equivalent definitions:~~
	*A group is residually finite if for ~~each non-identity element in the group, there is a [[normal subgroup]] of finite [[index (group theory)\|index]] not containing that element~~.
	*A group is residually finite if and only if the intersection of all its subgroups of finite index is trivial.
	*A group is residually finite if and only if the intersection of all its normal subgroups of finite index is trivial.
	*A group is residually finite if and only if it can be embedded inside the [[direct product of groups\|direct product]] of a ~~family of finite groups.~~

	~~== Examples ==~~

	~~Examples of groups that are residually finite are [[finite group]]s, [[free group]]s, [[finitely generated group\|finitely generated]] [[nilpotent group]]s, [[polycyclic-by-finite group]]~~s~~, [[finitely generated group\|finitely generated]] [[General linear group\|linear groups]], and [[fundamental group]]s of [[3-manifold]]s.~~

	Subgroups of residually finite groups are residually finite, and direct products of residually finite groups are residually finite. Any [[inverse limit]] of residually finite groups is residually finite. In particular, all [[profinite group]]s are residually finite.

	~~== Profinite topology ==~~

	~~Every group ''G'' may be made into a [[topological group]] by taking as a basis of open neighbourhoods of the identity, the collection of all normal subgroups of finite index~~ in ~~''G''~~. ~~The resulting [[topology]] is called the [[profinite group\|profinite topology]] on ''G''. A group is residually finite if, and only if, its profinite topology is [[Hausdorff space\|Hausdorff]].~~

	~~A group whose cyclic subgroups are closed in the profinite topology is said to be~~ <~~math~~>~~\Pi_C\,~~<~~/math~~>.
	~~Groups, each of whose finitely generated subgroups are closed in the profinite topology are called '''subgroup separable''' (also '''LERF''', for ''locally extended residually finite'').~~
	~~A group in which every [[conjugacy class]] is closed in the profinite topology is called '''conjugacy separable'''.~~

	~~== Varieties of residually finite groups ==~~

	~~One question is~~: ~~what are the properties of a [[variety (universal algebra)\|variety]] all of whose groups are residually finite? Two results about these are:~~

	* Any variety comprising only residually finite groups is generated by an [[A-group]].
	* For any variety comprising only residually finite groups, it contains a finite group such that all members are embedded in a direct product of that finite group.

	~~== See also ==~~
	* [[Residual property (mathematics)]]

	~~==External links==~~
	* [http://www.~~turpion~~.~~org~~/~~php~~/~~full/infoFT.phtml?journal_id=im&paper_id=807&year_id=1969&volume_id=3&issue_id=4&fpage=867 Article with proof of some of the above statements]~~

	~~[[Category:Infinite group theory]]~~
	~~[[Category:Properties of groups]~~]

en>Myasuda: /* References */ added missing diacritics

2013-02-16T14:51:25Z

References: added missing diacritics

← Older revision		Revision as of 16:51, 16 February 2013
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	~~Oscar~~ is ~~what my spouse loves~~ to ~~contact me and I completely dig~~ that ~~name~~. ~~Years~~ in the ~~past we moved to Puerto Rico~~ and ~~my family loves it~~. ~~To collect coins~~ is ~~what his~~ family and ~~him appreciate~~. ~~My working day occupation~~ is a ~~meter reader~~.<br><br>~~Stop~~ by ~~my site .~~.. [http://www.~~adosphere~~.~~com~~/~~poyocum adosphere~~.~~com~~]		In the [[mathematics\|mathematical]] field of [[group theory]], a [[Group (mathematics)\|group]] ''G'' is '''residually finite''' or '''finitely approximable''' if for every nontrivial element ''g'' in ''G'' there is a [[Group homomorphism\|homomorphism]] ''h'' from ''G'' to a finite group, such that

			:<math>h(g) \neq 1.\,</math>

			There are a number of equivalent definitions:
			*A group is residually finite if for each non-identity element in the group, there is a [[normal subgroup]] of finite [[index (group theory)\|index]] not containing that element.
			*A group is residually finite if and only if the intersection of all its subgroups of finite index is trivial.
			*A group is residually finite if and only if the intersection of all its normal subgroups of finite index is trivial.
			*A group is residually finite if and only if it can be embedded inside the [[direct product of groups\|direct product]] of a family of finite groups.

			== Examples ==

			Examples of groups that are residually finite are [[finite group]]s, [[free group]]s, [[finitely generated group\|finitely generated]] [[nilpotent group]]s, [[polycyclic-by-finite group]]s, [[finitely generated group\|finitely generated]] [[General linear group\|linear groups]], and [[fundamental group]]s of [[3-manifold]]s.

			Subgroups of residually finite groups are residually finite, and direct products of residually finite groups are residually finite. Any [[inverse limit]] of residually finite groups is residually finite. In particular, all [[profinite group]]s are residually finite.

			== Profinite topology ==

			Every group ''G'' may be made into a [[topological group]] by taking as a basis of open neighbourhoods of the identity, the collection of all normal subgroups of finite index in ''G''. The resulting [[topology]] is called the [[profinite group\|profinite topology]] on ''G''. A group is residually finite if, and only if, its profinite topology is [[Hausdorff space\|Hausdorff]].

			A group whose cyclic subgroups are closed in the profinite topology is said to be <math>\Pi_C\,</math>.
			Groups, each of whose finitely generated subgroups are closed in the profinite topology are called '''subgroup separable''' (also '''LERF''', for ''locally extended residually finite'').
			A group in which every [[conjugacy class]] is closed in the profinite topology is called '''conjugacy separable'''.

			== Varieties of residually finite groups ==

			One question is: what are the properties of a [[variety (universal algebra)\|variety]] all of whose groups are residually finite? Two results about these are:

			* Any variety comprising only residually finite groups is generated by an [[A-group]].
			* For any variety comprising only residually finite groups, it contains a finite group such that all members are embedded in a direct product of that finite group.

			== See also ==
			* [[Residual property (mathematics)]]

			==External links==
			* [http://www.turpion.org/php/full/infoFT.phtml?journal_id=im&paper_id=807&year_id=1969&volume_id=3&issue_id=4&fpage=867 Article with proof of some of the above statements]

			[[Category:Infinite group theory]]
			[[Category:Properties of groups]]

en>Rjwilmsi: /* References */Journal cites:, added 1 DOI, using AWB (7751)

2011-07-23T09:52:41Z

References: Journal cites:, added 1 DOI, using AWB (7751)

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Oscar is what my spouse loves to contact me and I completely dig that name. Years in the past we moved to Puerto Rico and my family loves it. To collect coins is what his family and him appreciate. My working day occupation is a meter reader.<br><br>Stop by my site ... [http://www.adosphere.com/poyocum adosphere.com]