Difference between revisions of "Group"
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* There exists an element <math>e\in G</math> so that for all <math>g\in G</math>, <math>ge=eg=g</math> ([[identity]]). | * There exists an element <math>e\in G</math> so that for all <math>g\in G</math>, <math>ge=eg=g</math> ([[identity]]). | ||
* For any <math>g\in G</math>, there exists <math>g^{-1}\in G</math> so that <math>gg^{-1}=g^{-1}g=e</math> ([[Inverse with respect to an operation | inverses]]). | * For any <math>g\in G</math>, there exists <math>g^{-1}\in G</math> so that <math>gg^{-1}=g^{-1}g=e</math> ([[Inverse with respect to an operation | inverses]]). | ||
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| + | Note that the group operation need not be [[commutative]]. If the group operation is commutative, we call the group an [[abelian group]] (after the Norwegian mathematician Niels Henrik Abel). | ||
Groups frequently arise as [[permutation]]s of collections of objects. For example, the rigid motions of <math>\mathbb{R}^2</math> that fix a certain regular <math>n</math>-gon is a group, called the [[dihedral group]] and denoted <math>D_{2n}</math> (since it has <math>2n</math> elements). Another example of a group is the [[symmetric group]] <math>S_n</math> of all permutations of <math>\{1,2,\ldots,n\}</math>. | Groups frequently arise as [[permutation]]s of collections of objects. For example, the rigid motions of <math>\mathbb{R}^2</math> that fix a certain regular <math>n</math>-gon is a group, called the [[dihedral group]] and denoted <math>D_{2n}</math> (since it has <math>2n</math> elements). Another example of a group is the [[symmetric group]] <math>S_n</math> of all permutations of <math>\{1,2,\ldots,n\}</math>. | ||
Related algebraic structures are [[ring]]s and [[field]]s. | Related algebraic structures are [[ring]]s and [[field]]s. | ||
Revision as of 09:21, 18 July 2006
This article is a stub. Help us out by expanding it.
A group 
is a set of elements together with an operation 
(the dot is frequently supressed) satisfying the following conditions:
- For all
, 
(associativity). - There exists an element
so that for all 
, 
(identity). - For any , there exists
so that 
( inverses).
, 
(
so that for all 
, 
(
so that 
(Note that the group operation need not be commutative. If the group operation is commutative, we call the group an abelian group (after the Norwegian mathematician Niels Henrik Abel).
Groups frequently arise as permutations of collections of objects. For example, the rigid motions of 
that fix a certain regular 
-gon is a group, called the dihedral group and denoted 
(since it has 
elements). Another example of a group is the symmetric group 
of all permutations of 
.
