Using that a subgroup is normal - Revision history

Vipul at 01:37, 20 February 2013

2013-02-20T01:37:09Z

← Older revision		Revision as of 01:37, 20 February 2013
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	* The order of the [[quotient group]] equals the index of the normal subgroup (this is a corollary of [[Lagrange's theorem]]). This puts constraints on the nature of the quotient group based on information about the normal subgroup.		* The order of the [[quotient group]] equals the index of the normal subgroup (this is a corollary of [[Lagrange's theorem]]). This puts constraints on the nature of the quotient group based on information about the normal subgroup.
	* If the normal subgroup has a [[permutable complements\|permutable complement]] in the whole group, [[complement to normal subgroup is isomorphic to quotient group\|it is isomorphic to the quotient group]].		* If the normal subgroup has a [[permutable complements\|permutable complement]] in the whole group (i.e., it is a [[complemented normal subgroup]]), [[complement to normal subgroup is isomorphic to quotient group\|it is isomorphic to the quotient group]].

			==Using the isomorphism theorems==

			Normal subgroups set the stage for using the isomorphism theorems. The major ones are listed below.

			{\| class="sortable" border="1"
			! Theorem name/number !! Statement
			\|-
			\| [[First isomorphism theorem]] \|\| If <math>\varphi:G \to H</math> is a surjective homomorphism, then the kernel <math>N</math> of <math>\varphi</math> is a normal subgroup, and if <math>\alpha:G \to G/N</math> is the quotient map, then there is a unique isomorphism <math>\psi:G/N \to H</math> such that <math>\psi \circ \alpha = \varphi</math>.
			\|-
			\| [[Second isomorphism theorem]] (diamond isomorphism theorem) \|\| If <math>N,H \le G</math> such that <math>N</math> is contained in the normalizer of <math>H</math>, then <math>N</math> is normal in <math>NH</math> and <math>NH/N \cong H/(H \cap N)</math>.
			\|-
			\| [[Third isomorphism theorem]] \|\| If <math>H \le K \le G</math> with both <math>H,K</math> normal in <math>G</math>, then <math>H</math> is normal in <math>K</math> and <math>(G/H)/(K/H) \cong G/K</math>.
			\|-
			\| [[Fourth isomorphism theorem]] (lattice isomorphism theorem) \|\| If <math>H</math> is normal in <math>G</math>, there is a bijective correspondence between subgroups of <math>G/H</math> and subgroups of <math>G</math> containing <math>H</math>, satisfying many nice conditions.
			\|}

Vipul at 01:34, 20 February 2013

2013-02-20T01:34:04Z

← Older revision		Revision as of 01:34, 20 February 2013
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	This article explores the various ways in which, given a group and a subgroup (through some kind of description) we can ''use'' the fact that the subgroup is a [[normal subgroup]].		This article explores the various ways in which, given a group and a subgroup (through some kind of description) we can ''use'' the fact that the subgroup is a [[normal subgroup]].

			==Ability to take the quotient group==

			One of the main advantages of working with normal subgroups is that we can take the [[quotient group]] of the whole group by the subgroup. Explicitly, if <math>H</math> is a normal subgroup of a group <math>G</math>, we can consider the quotient group <math>G/H</math>. This has a number of applications:

			* We might be able to use facts that we know about <math>H</math> to deduce constraints on <math>G/H</math>.
			* We might be able to deduce facts about <math>G</math> from facts that we know about <math>H</math> and that we deduce about <math>G/H</math>.
			* This might be the template for the induction step in a proof by induction. See [[induction for finite groups]] for more.

			===Examples of deducing the structure of the quotient group===

			We have the following structural restrictions on the quotient group:

			* The order of the [[quotient group]] equals the index of the normal subgroup (this is a corollary of [[Lagrange's theorem]]). This puts constraints on the nature of the quotient group based on information about the normal subgroup.
			* If the normal subgroup has a [[permutable complements\|permutable complement]] in the whole group, [[complement to normal subgroup is isomorphic to quotient group\|it is isomorphic to the quotient group]].

Vipul: Vipul moved page Using normality to Using that a subgroup is normal over redirect

2013-02-20T01:22:12Z

Vipul moved page Using normality to Using that a subgroup is normal over redirect

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Vipul: Created page with "{{property satisfaction use technique survey article|normal subgroup}} This article explores the various ways in which, given a group and a subgroup (through some kind of descri..."

2011-04-20T00:53:25Z

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{{property satisfaction use technique survey article|normal subgroup}}

This article explores the various ways in which, given a group and a subgroup (through some kind of description) we can ''use'' the fact that the subgroup is a [[normal subgroup]].