Absolutely simple group - Revision history

en>K9re11: removed Category:Group theory using HotCat as there is already the more specific category ''properties of groups''

2014-11-13T13:53:23Z

removed Category:Group theory using HotCat as there is already the more specific category ''properties of groups''

← Older revision		Revision as of 15:53, 13 November 2014
Line 1:		Line 1:
	~~In [[mathematics]]~~, ~~in the field of [[group theory]], a [[subgroup]] of a [[group (mathematics)\|group]]~~ is ~~termed~~ a ~~[[retract]] if there is an [[endomorphism]] of the group that maps [[surjective]]ly to the subgroup~~ and is ~~identity on the subgroup~~. ~~In symbols,~~ <~~math~~>H</~~math> is a retract of <math>G<~~/~~math> if and only if there is an endomorphism <math>\sigma:G \to G<~~/~~math> such that <math>\sigma(h) = h<~~/~~math> for all <math>h \in H<~~/~~math> and <math>\sigma(g) \in H<~~/~~math> for all <math>g \in G<~~/~~math>~~.		Nice to meet you, my title is Araceli Oquendo but I don't like when people use my full title. Alabama has always been his home and his family enjoys it. I am a production and distribution officer. Playing croquet is some thing I will by no means give up.<br><br>Review my blog :: extended car warranty ([http://Bencorpusa.com/ActivityFeed/MyProfile/tabid/127/userId/152431/Default.aspx click through the next internet site])

	~~The endomorphism itself is termed an [[idempotent]] endomorphism or a retraction.~~

	~~The following is known about retracts:~~

	* A subgroup is a retract if and only if it has a [[normal subgroup\|normal]~~] [[complement (group theory~~)~~\|complement]]. The normal complement, specifically, is the kernel of the retraction.~~
	* Every [[direct product of groups\|direct factor]] is a retract. Conversely, any retract which is a normal subgroup is a direct factor.
	* Every retract has the [[CEP subgroup\|congruence extension property]].
	* Every [[regular factor]], and in particular, every [[free factor]], is a retract.

	~~==References==~~
	~~{{unreferenced\|date=September 2008}}~~

	~~[[Category:Group theory]]~~
	~~[[Category:Subgroup properties]]~~


	~~{{Abstract-algebra-stub}}~~

en>AnomieBOT: Dating maintenance tags: {{Unreferenced}}

2013-12-22T05:14:04Z

Dating maintenance tags: {{Unreferenced}}

← Older revision		Revision as of 07:14, 22 December 2013
Line 1:		Line 1:
	~~Claude~~ is ~~her title and she~~ [~~http://Autos.Jdpower.com/content/buying-tip/xa345A7/pros-and-cons-of-buying-~~an~~-extended-warranty-for-a-car.htm completely digs~~] that ~~name~~. ~~Alabama~~ is ~~where he~~ and ~~his wife live and [http~~://~~www.tor-und-fenstertechnik.de~~/~~index~~/~~users~~/~~view/id/20377 extended car warranty~~] ~~he has every thing that he requirements there~~. ~~Meter reading~~ is ~~where my primary earnings comes from~~ [~~http://Www~~.~~Daveramsey~~.~~com/article/just-say-no-to-extended-warranties-dr/lifeandmoney_automobiles/ auto warranty~~] ~~but soon I'll be on my personal~~. ~~What she enjoys doing~~ is ~~to perform~~ [~~http://ganafc~~.~~com/xe/link/2514287 extended car warranty~~] ~~croquet but she hasn't~~ [~~https://Autoclubsouth~~.~~aaa.com/Automotive/auto_warranty.aspx?nvbar~~=~~Automotive~~:~~ExtendedWarranty produced~~] ~~a dime with it.<br><br>Also visit my page - car warranty (~~[~~http~~:~~//www.Gettingtherefromhere.info/User~~-~~Profile/userId/13851 simply click the following webpage])~~		In [[mathematics]], in the field of [[group theory]], a [[subgroup]] of a [[group (mathematics)\|group]] is termed a [[retract]] if there is an [[endomorphism]] of the group that maps [[surjective]]ly to the subgroup and is identity on the subgroup. In symbols, <math>H</math> is a retract of <math>G</math> if and only if there is an endomorphism <math>\sigma:G \to G</math> such that <math>\sigma(h) = h</math> for all <math>h \in H</math> and <math>\sigma(g) \in H</math> for all <math>g \in G</math>.

			The endomorphism itself is termed an [[idempotent]] endomorphism or a retraction.

			The following is known about retracts:

			* A subgroup is a retract if and only if it has a [[normal subgroup\|normal]] [[complement (group theory)\|complement]]. The normal complement, specifically, is the kernel of the retraction.
			* Every [[direct product of groups\|direct factor]] is a retract. Conversely, any retract which is a normal subgroup is a direct factor.
			* Every retract has the [[CEP subgroup\|congruence extension property]].
			* Every [[regular factor]], and in particular, every [[free factor]], is a retract.

			==References==
			{{unreferenced\|date=September 2008}}

			[[Category:Group theory]]
			[[Category:Subgroup properties]]


			{{Abstract-algebra-stub}}

en>Qetuth: more specific stub type

2012-01-01T10:44:25Z

more specific stub type

New page

Claude is her title and she [http://Autos.Jdpower.com/content/buying-tip/xa345A7/pros-and-cons-of-buying-an-extended-warranty-for-a-car.htm completely digs] that name. Alabama is where he and his wife live and [http://www.tor-und-fenstertechnik.de/index/users/view/id/20377 extended car warranty] he has every thing that he requirements there. Meter reading is where my primary earnings comes from [http://Www.Daveramsey.com/article/just-say-no-to-extended-warranties-dr/lifeandmoney_automobiles/ auto warranty] but soon I'll be on my personal. What she enjoys doing is to perform [http://ganafc.com/xe/link/2514287 extended car warranty] croquet but she hasn't [https://Autoclubsouth.aaa.com/Automotive/auto_warranty.aspx?nvbar=Automotive:ExtendedWarranty produced] a dime with it.<br><br>Also visit my page - car warranty ([http://www.Gettingtherefromhere.info/User-Profile/userId/13851 simply click the following webpage])