Difference between revisions of "Group"
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| − | A '''group''' <math>G</math> is a set of elements together with an operation <math>\cdot:G\times G\to G</math> (the dot is frequently supressed) satisfying the following conditions: | + | {{stub}} |
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| + | A '''group''' <math>G</math> is a [[set]] of elements together with an [[operation]] <math>\cdot:G\times G\to G</math> (the dot is frequently supressed) satisfying the following conditions: | ||
* For all <math>a,b,c\in G</math>, <math>a(bc)=(ab)c</math> ([[associative|associativity]]). | * For all <math>a,b,c\in G</math>, <math>a(bc)=(ab)c</math> ([[associative|associativity]]). | ||
* 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> ([[ | + | * 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]]). |
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. | ||
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Revision as of 09:19, 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 
(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 
.
