Quotient map - Revision history

Vipul at 14:55, 26 June 2008

2008-06-26T14:55:55Z

← Older revision		Revision as of 14:55, 26 June 2008
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	===Definition with symbols===		===Definition with symbols===

	Let <math>G</math> be a [[group]] and <math>N</math> be a [[normal subgroup]]. Consider the map <math>\~~phi~~:G \to G/N</math> which sends <math>g \in G</math> to the coset <math>gN</math>. This map is termed the '''quotient map'''.		Let <math>G</math> be a [[group]] and <math>N</math> be a [[normal subgroup]]. Consider the map <math>\varphi:G \to G/N</math> which sends <math>g \in G</math> to the coset <math>gN</math>. This map is termed the '''quotient map'''.

	The equivalence relation induced by the cosets of a normal subgroup is a congruence, viz:		The equivalence relation induced by the cosets of a normal subgroup is a congruence, viz.:

	* <math>aN = bN, cN = dN \implies acN = bdN</math>		* <math>aN = bN, cN = dN \implies acN = bdN</math>
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	The group <math>G/N</math> is also termed the '''quotient group''' of <math>N</math> via this quotient map.		The group <math>G/N</math> is also termed the '''quotient group''' of <math>N</math> via this quotient map.

	==~~External links~~==		==Related facts==

	~~===Discussions/Motivational material===~~		* [[Normal subgroup equals kernel of homomorphism]]: The kernel of any homomorphism is a normal subgroup. Conversely, any normal subgroup is a kernel of a homomorphism, namely the quotient map for that normal subgroup.
			* [[First isomorphism theorem]]: This states that for any [[homomorphism of groups]] <math>\varphi:G \to H</math> with kernel <math>N</math>, the map from <math>G</math>, restricted to the image, is equivalent to the quotient map <math>G \to G/N</math>
	* [~~http~~://~~polymathematics.typepad.com~~/~~polymath~~/~~2006~~/~~10/weeks_7_and_8.html Polymathematics discussion on good examples for normal subgroups~~]		* [[Second isomorphism theorem]]
			* [[Third isomorphism theorem]]
			* [[Fourth isomorphism theorem]]

Vipul: 2 revisions

2008-05-08T00:06:48Z

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Vipul at 05:55, 27 March 2007

2007-03-27T05:55:38Z

← Older revision		Revision as of 05:55, 27 March 2007
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	The group <math>G/N</math> is also termed the '''quotient group''' of <math>N</math> via this quotient map.		The group <math>G/N</math> is also termed the '''quotient group''' of <math>N</math> via this quotient map.

			==External links==

			===Discussions/Motivational material===

			* [http://polymathematics.typepad.com/polymath/2006/10/weeks_7_and_8.html Polymathematics discussion on good examples for normal subgroups]

Vipul: Started the page

2007-03-15T00:01:43Z

Started the page

New page

{{basicdef}}

==Definition==

===Symbol-free definition===

Given a [[group]] and a [[normal subgroup]], the '''quotient map''' corresponding to that normal subgroup is the map from the group to the coset space (that is, the set of cosets) of this normal subgroup.

Since the equivalence relation induced by ''being in the same coset'' of the normal subgroup is a [[congruence on a group|congruence]], the coset space can actually be equipped with a canonical group structure, and the quotient map then becomes a surjective group homomorphism.

The coset space with this group structure is also termed the '''quotient group'''.

===Definition with symbols===

Let <math>G</math> be a [[group]] and <math>N</math> be a [[normal subgroup]]. Consider the map <math>\phi:G \to G/N</math> which sends <math>g \in G</math> to the coset <math>gN</math>. This map is termed the '''quotient map'''.

The equivalence relation induced by the cosets of a normal subgroup is a congruence, viz:

* <math>aN = bN, cN = dN \implies acN = bdN</math>
* <math>aN = bN \implies a^{-1}N = b^{-1}N</math>

Thus, we can define a group structure on the coset space by setting <math>(aN)(bN) = abN</math> (this is well-defined and independent of the choice of coset representatives precisely because of the above fact).

Note that the quotient map is a surjective homomorphism whose kernel is the given normal subgroup.

The group <math>G/N</math> is also termed the '''quotient group''' of <math>N</math> via this quotient map.