Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Random
 sections EncyclopædiaPapersBooksExpositionsmeta Requests (72)Orphanage (11)Unclass'd (1)Unproven (264)Corrections (122)talkback PollsForumsFeedbackBug Reportsdownloads SnapshotsPM Bookinformation DocsClassificationNewsLegaleseHistoryChangeLogTODO List
 group (Definition)
 Group. A group is a pair where is a non-empty set and is binary operation on that holds the following conditions. For any , . (Associativity of the operation). For any in , belongs to . (The operation is closed). There is an element such that for any . (Existence of identity element). For any there exists an element such that . (Existence of inverses). Usually the symbol is omitted and we write for . Sometimes, the symbol is used to represent the operation, especially when the group is abelian. It can be proved that there is only one identity element , and that for every element there is only one inverse. Because of this we usually denote the inverse of as or when we are using additive notation. The identity element is also called neutral element due to its behavior with respect to the operation. Groups often arise as the symmetry groups of other mathematical objects; the study of such situations uses group actions. In fact, much of the study of groups themselves is conducted using group actions.

"group" is owned by drini. [ full author list (2) ]
(view preamble)

 View style: HTML with imagespage imagesTeX source

See Also: subgroup, cyclic group, simple group, symmetric group, free group, ring, field, group homomorphism, Lagrange's theorem, identity element, proper subgroup, groupoid, fundamental group, topological group, Lie group, Proof: The orbit of any element of a group is a subgroup, locally cyclic group, existence of Hilbert class field, abelian group, , example of fibre product, group object, group scheme, subring, group action, general linear group

 Also defines: inverse, identity
 Keywords: ring, algebra, morphism, subgroup, group, set

 Attachments: identity element (Definition) by mclase examples of groups (Example) by AxelBoldt uniqueness of inverse (for groups) (Result) by waj

Cross-references: group actions, symmetry, additive, abelian, represent, identity element, closed, associativity, binary operation
There are 398 references to this object.

This is version 13 of group, born on 2001-08-29, modified 2004-10-02.
Object id is 78, canonical name is Group.
Accessed 17167 times total.

Classification:
 AMS MSC: 20-00 (Group theory and generalizations :: General reference works ) 20A05 (Group theory and generalizations :: Foundations :: Axiomatics and elementary properties)