Group - Revision history

Vipul: 3 revisions

2008-05-18T19:46:43Z

3 revisions

← Older revision	Revision as of 19:46, 18 May 2008
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Vipul: /* External links */

2007-12-13T20:37:14Z

External links

← Older revision		Revision as of 20:37, 13 December 2007
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	==External links==		==External links==

			===Definition links===

	* {{gp\|Group}}		* {{gp\|Group}}
	* {{wp\|Group}}		* {{wp\|Group}}
			* {{cz\|Group (mathematics)}}

Vipul at 20:35, 13 December 2007

2007-12-13T20:35:58Z

← Older revision		Revision as of 20:35, 13 December 2007
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			{{groupprops version at\|Group}}

	==Definition==		==Definition==

Vipul at 08:12, 27 August 2007

2007-08-27T08:12:27Z

New page

==Definition==

A '''group''' is a set <math>G</math> equipped with three additional operations:

* A binary operation <math>*</math> called multiplication, or product
* A unary operation denoted as <math>g \mapsto g^{-1}</math> called the inverse map
* A constant element denoted <math>e</math>

such that the following conditions hold:

* <math>a * (b * c) = (a * b) * c \ \forall \ a, b, c \in G</math>
* <math>a * e = e * a = a \ \forall \ a \in G</math>
* <math>a * a^{-1} = a^{-1} * a = e \ \forall \ a \in G</math>

==Importance==

Groups arise in differential geometry, primarily in the following contexts:

* As symmetries, or automorphisms, of geometric structures
* As structure groups of bundles
* As fundamental groups or higher (co)homotopy and (co)homology groups
* As manifolds themselves. Notions of relevance here are [[topological group]] and [[Lie group]]

==External links==

* {{gp|Group}}
* {{wp|Group}}