Group - Revision history

Vipul at 14:57, 29 December 2009

2009-12-29T14:57:37Z

← Older revision		Revision as of 14:57, 29 December 2009
Line 5:		Line 5:
	A '''group''' is a [[small category]] with one object where all morphisms are isomorphisms. (note that the ''small'' assumption is to ensure that the morphisms form a set; however, it can be dropped for some purposes).		A '''group''' is a [[small category]] with one object where all morphisms are isomorphisms. (note that the ''small'' assumption is to ensure that the morphisms form a set; however, it can be dropped for some purposes).

	===~~Ordinary~~ definition===		===Universal algebra definition===

	A '''group''' is a set <math>G</math> equipped with a binary operation <math>*</math>, a unary operation <math>{}^{-1}</math>, and a constant <math>e</math> such that the following hold:		A '''group''' is a set <math>G</math> equipped with a binary operation <math>*</math>, a unary operation <math>{}^{-1}</math>, and a constant <math>e</math> such that the following hold:
Line 24:		Line 24:

	Further, this equivalence gives an equivalence of categories, where the first category is the subcategory corresponding to groups of the [[category of locally small categories]] (so the morphisms are [[functor]]s), and the second category is the [[category of groups]] in the conventional sense.		Further, this equivalence gives an equivalence of categories, where the first category is the subcategory corresponding to groups of the [[category of locally small categories]] (so the morphisms are [[functor]]s), and the second category is the [[category of groups]] in the conventional sense.

			==Category of groups==

			The collection of groups, with a suitably defined notion of homomorphism, forms a category called the [[category of groups]]. This is a [[concrete category]], and corresponds to the standard way of obtaining a category from the universal algebra definition of group.

	==External links==		==External links==

Vipul: /* External links */

2009-12-29T13:19:49Z

External links

← Older revision		Revision as of 13:19, 29 December 2009
Line 34:		Line 34:
	===Other resources===		===Other resources===

	* [[nlab:group\|~~group~~ on nlab]]		* [[nlab:group\|Group on nlab]]
	* [[Wikipedia:Group\|Group on Wikipedia]]		* [[Wikipedia:Group\|Group on Wikipedia]]
			* [[Mathworld:Group\|Group on Wolfram Mathworld]]
			* [[Planetmath:Group\|Group on Planetmath]]

Vipul: /* Other resources */

2009-12-29T13:01:09Z

Other resources

← Older revision		Revision as of 13:01, 29 December 2009
Line 36:		Line 36:
	* [[nlab:group\|group on nlab]]		* [[nlab:group\|group on nlab]]
	* [[Wikipedia:Group\|Group on Wikipedia]]		* [[Wikipedia:Group\|Group on Wikipedia]]
	* [[Pm:Group\|Group on Planetmath]]
	* [[Mw:Group\|Group on Mathworld]]

Vipul: /* Other resources */

2009-12-29T13:00:38Z

Other resources

← Older revision		Revision as of 13:00, 29 December 2009
Line 36:		Line 36:
	* [[nlab:group\|group on nlab]]		* [[nlab:group\|group on nlab]]
	* [[Wikipedia:Group\|Group on Wikipedia]]		* [[Wikipedia:Group\|Group on Wikipedia]]
	* [[~~Planetmath~~:Group\|Group on Planetmath]]		* [[Pm:Group\|Group on Planetmath]]
	* [[~~Mathworld~~:Group\|Group on Mathworld]]		* [[Mw:Group\|Group on Mathworld]]

Vipul: /* External links */

2009-12-29T13:00:10Z

External links

← Older revision		Revision as of 13:00, 29 December 2009
Line 27:		Line 27:
	==External links==		==External links==

	===Other subject ~~wiki links~~===		===Other subject wikis===

	* [[Groupprops:Group\|Group on the Group Properties Wiki]] -- the most detailed and canonical reference.		* [[Groupprops:Group\|Group on the Group Properties Wiki]] -- the most detailed and canonical reference.
	* [[Topospaces:Group\|Group on the Topology Wiki]]		* [[Topospaces:Group\|Group on the Topology Wiki]]

	===Other ~~links~~===		===Other resources===

	* [[nlab:group\|group on nlab]]		* [[nlab:group\|group on nlab]]
			* [[Wikipedia:Group\|Group on Wikipedia]]
			* [[Planetmath:Group\|Group on Planetmath]]
			* [[Mathworld:Group\|Group on Mathworld]]

Vipul at 12:58, 29 December 2009

2009-12-29T12:58:50Z

← Older revision		Revision as of 12:58, 29 December 2009
Line 14:		Line 14:

	(there are other slightly different formulations of this definition).		(there are other slightly different formulations of this definition).

			===Definition building on monoid===

			A group is a [[monoid]] in which every element has a two-sided inverse.

	===Equivalence of definitions===		===Equivalence of definitions===
Line 27:		Line 31:
	* [[Groupprops:Group\|Group on the Group Properties Wiki]] -- the most detailed and canonical reference.		* [[Groupprops:Group\|Group on the Group Properties Wiki]] -- the most detailed and canonical reference.
	* [[Topospaces:Group\|Group on the Topology Wiki]]		* [[Topospaces:Group\|Group on the Topology Wiki]]

			===Other links===

			* [[nlab:group\|group on nlab]]

Vipul: Created page with '==Definition== ===Category-theoretic definition=== A '''group''' is a small category with one object where all morphisms are isomorphisms. (note that the ''small'' assumpti…'

2009-12-29T12:51:43Z

Created page with '==Definition== ===Category-theoretic definition=== A '''group''' is a small category with one object where all morphisms are isomorphisms. (note that the ''small'' assumpti…'

New page

==Definition==

===Category-theoretic definition===

A '''group''' is a [[small category]] with one object where all morphisms are isomorphisms. (note that the ''small'' assumption is to ensure that the morphisms form a set; however, it can be dropped for some purposes).

===Ordinary definition===

A '''group''' is a set <math>G</math> equipped with a binary operation <math>*</math>, a unary operation <math>{}^{-1}</math>, and a constant <math>e</math> such that the following hold:

* '''Associativity''': <math>a * (b * c) = (a * b) * c \ \forall \ a,b,c \in G</math>.
* '''Neutral element''' or '''identity element''': <math>a * e = e * a = a \ \forall \ a \in G</math>.
* '''Inverse element''': <math>a * a^{-1} = a^{-1} * a = e</math>.

(there are other slightly different formulations of this definition).

===Equivalence of definitions===

The two definitions are equivalent in the following sense: given any group in the category-theoretic set, the morphisms, under composition, form a group in the set-theoretic sense. Conversely, given any group in the set-theoretic sense, we can construct a category with one object and with the morphisms corresponding to the elements of the group.

Further, this equivalence gives an equivalence of categories, where the first category is the subcategory corresponding to groups of the [[category of locally small categories]] (so the morphisms are [[functor]]s), and the second category is the [[category of groups]] in the conventional sense.

==External links==

===Other subject wiki links===

* [[Groupprops:Group|Group on the Group Properties Wiki]] -- the most detailed and canonical reference.
* [[Topospaces:Group|Group on the Topology Wiki]]

← Older revision		Revision as of 13:01, 29 December 2009
Line 36:		Line 36:
	* [[nlab:group\|group on nlab]]		* [[nlab:group\|group on nlab]]
	* [[Wikipedia:Group\|Group on Wikipedia]]		* [[Wikipedia:Group\|Group on Wikipedia]]
	* [[Pm:Group\|Group on Planetmath]]
	* [[Mw:Group\|Group on Mathworld]]

← Older revision		Revision as of 13:00, 29 December 2009
Line 36:		Line 36:
	* [[nlab:group\|group on nlab]]		* [[nlab:group\|group on nlab]]
	* [[Wikipedia:Group\|Group on Wikipedia]]		* [[Wikipedia:Group\|Group on Wikipedia]]
	* [[~~Planetmath~~:Group\|Group on Planetmath]]		* [[Pm:Group\|Group on Planetmath]]
	* [[~~Mathworld~~:Group\|Group on Mathworld]]		* [[Mw:Group\|Group on Mathworld]]