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group (Definition)

Group.
A group is a pair $ (G,*)$ where $ G$ is a non-empty set and $ *$ is binary operation on $ G$ that holds the following conditions.

  • For any $ a,b,c\in G$, $ (a*b)*c=a*(b*c)$. (Associativity of the operation).
  • For any $ a,b$ in $ G$, $ a*b$ belongs to $ G$. (The operation $ *$ is closed).
  • There is an element $ e\in G$ such that $ g*e=e*g=g$ for any $ g\in G$. (Existence of identity element).
  • For any $ g\in G$ there exists an element $ h$ such that $ g*h=h*g=e$. (Existence of inverses).

Usually the symbol $ *$ is omitted and we write $ ab$ for $ a*b$. 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 $ a$ as $ a^{-1}$ or $ -a$ 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) ]
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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, $\mathit{SL}_2(F_3)$, 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 MSC20-00 (Group theory and generalizations :: General reference works )
 20A05 (Group theory and generalizations :: Foundations :: Axiomatics and elementary properties)

Pending Errata and Addenda
None.
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Discussion
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I+ll write more later by drini on 2001-08-29 18:18:07
I dont have more time now...
I+ll add examples and stuff later
     f                     
G ---------> H              
\         ^     G          
p \       /_   -----  ~ f(G)
  \     / f   ker f        
   Y   /                   
  G/ker f                  
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