Normal-extensible automorphism - Revision history

Vipul: /* Facts */

2009-06-01T16:26:05Z

Facts

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	* [[Centerless and maximal in automorphism group implies every automorphism is normal-extensible]]		* [[Centerless and maximal in automorphism group implies every automorphism is normal-extensible]]
	* [[Every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible]]		* [[Every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible]]
			* [[NRPC conjecture implies normal-extensible equals characteristic-extensible]]

	More facts about characterizing normal-extensible automorphisms can be found at the [[normal-extensible automorphisms problem]] page.		More facts about characterizing normal-extensible automorphisms can be found at the [[normal-extensible automorphisms problem]] page.

	==Metaproperties==		==Metaproperties==

2009-05-31T13:49:40Z

Related group properties

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	* [[Group in which every automorphism is normal-extensible]]		* [[Group in which every automorphism is normal-extensible]]
			* [[Group in which every normal-extensible automorphism is inner]]

	==Facts==		==Facts==

2009-05-31T13:41:26Z

Relation with other properties

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	* [[Stronger than::Characteristic-extensible automorphism]]		* [[Stronger than::Characteristic-extensible automorphism]]
	* [[Stronger than::Central factor-extensible automorphism]]		* [[Stronger than::Central factor-extensible automorphism]]

			===Incomparable properties===

			* [[Normal automorphism]]: {{proofat\|[[Normal not implies normal-extensible]], [[normal-extensible not implies normal]]}}

	===Related group properties===		===Related group properties===

2009-05-31T13:40:09Z

← Older revision		Revision as of 13:40, 31 May 2009
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	* [[Group in which every automorphism is normal-extensible]]		* [[Group in which every automorphism is normal-extensible]]

			==Facts==

			* [[Centerless and maximal in automorphism group implies every automorphism is normal-extensible]]
			* [[Every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible]]

			More facts about characterizing normal-extensible automorphisms can be found at the [[normal-extensible automorphisms problem]] page.
	==Metaproperties==		==Metaproperties==

2009-04-03T20:02:44Z

Relation with other properties

← Older revision		Revision as of 20:02, 3 April 2009
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	===Stronger properties===		===Stronger properties===

	* [[Inner automorphism]]		* [[Weaker than::Extensible automorphism]]: {{proofofstrictimplicationat\|[[extensible implies normal-extensible]]\|[[normal-extensible not implies extensible]]}}. Also related:
	* [[~~Extensible~~ automorphism]]		** [[Weaker than::Inner automorphism]]
			** [[Weaker than::Subnormal-extensible automorphism]]
	* [[Iteratively normal-extensible automorphism]] viz <math>\alpha</math>-normal extensible automorphism for an ordinal <math>\alpha</math> that is at least 1		* [[Iteratively normal-extensible automorphism]] viz <math>\alpha</math>-normal extensible automorphism for an ordinal <math>\alpha</math> that is at least 1
	* [[Subnormal-extensible automorphism]]
	* [[Normal series-extensible automorphism]]		* [[Normal series-extensible automorphism]]

	===Weaker properties===		===Weaker properties===

	* [[Characteristic-extensible automorphism]]		* [[Stronger than::Characteristic-extensible automorphism]]
	* [[Central factor-extensible automorphism]]		* [[Stronger than::Central factor-extensible automorphism]]

	===Related group properties===		===Related group properties===

2008-08-12T20:54:56Z

← Older revision		Revision as of 20:54, 12 August 2008
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	* [[Characteristic-extensible automorphism]]		* [[Characteristic-extensible automorphism]]
	* [[Central factor-extensible automorphism]]		* [[Central factor-extensible automorphism]]

			===Related group properties===

			* [[Group in which every automorphism is normal-extensible]]

	==Metaproperties==		==Metaproperties==

2008-06-24T15:30:23Z

← Older revision		Revision as of 15:30, 24 June 2008
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			{{wikilocal}}
	{{automorphism property}}		{{automorphism property}}
			{{variation of\|extensible automorphism}}
	{{~~variationof~~\|extensible automorphism}}		{{term related to\|Extensible automorphisms problem}}

	{{~~termrelatedto~~\|Extensible automorphisms problem}}

	~~{{wikilocal~~}}

	==Definition==		==Definition==

2008-05-07T23:52:16Z

3 revisions

← Older revision	Revision as of 23:52, 7 May 2008
(No difference)

2007-07-21T02:33:12Z

In terms of the qualified extensibility operator

← Older revision		Revision as of 02:33, 21 July 2007
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	An automorphism <math>\sigma</math> of a group <math>G</math> is termed '''normal-extensible''' if, for any embedding of <math>G</math> as a normal subgroup of another group <math>H</math> there is an automorphism <math>\sigma'</math> of <math>H</math> such that the restriction of <math>\sigma'</math> to <math>G</math> is <math>\sigma</math>.		An automorphism <math>\sigma</math> of a group <math>G</math> is termed '''normal-extensible''' if, for any embedding of <math>G</math> as a normal subgroup of another group <math>H</math> there is an automorphism <math>\sigma'</math> of <math>H</math> such that the restriction of <math>\sigma'</math> to <math>G</math> is <math>\sigma</math>.

	===~~In terms of the~~ qualified extensibility operator~~===~~		==Formalisms==

			{{obtainedbyapplyingthe\|qualified extensibility operator\|normality}}

	The property of normal-extensibility arises by applying the [[qualified extensibility operator]] with the ''qualifying property'' being [[normality]] and the the automorphism property being the [[tautology]] (that is, the property of being ''any'' automorphism).		The property of normal-extensibility arises by applying the [[qualified extensibility operator]] with the ''qualifying property'' being [[normality]] and the the automorphism property being the [[tautology]] (that is, the property of being ''any'' automorphism).

2007-07-21T02:32:15Z

← Older revision		Revision as of 02:32, 21 July 2007
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	{{automorphism property}}		{{automorphism property}}

	{{termrelatedto\|Extensible ~~Automorphisms Problem~~}}		{{variationof\|extensible automorphism}}

			{{termrelatedto\|Extensible automorphisms problem}}

	{{wikilocal}}		{{wikilocal}}