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: | + | 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, so <math>ab</math> is written instead of <math>a\cdot b</math>) 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]]). | ||
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* 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]]). | ||
| − | + | (Equivalently, a group is a [[monoid]]s with 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). | 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 in some texts <math>D_{2n}</math> (since it has <math>2n</math> elements) and in others <math>D_n</math> (since it preserves a regular <math>n</math>-gon). 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 or symmetries 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 in some texts <math>D_{2n}</math> (since it has <math>2n</math> elements) and in others <math>D_n</math> (since it preserves a regular <math>n</math>-gon). Another example of a group is the [[symmetric group]] <math>S_n</math> of all permutations of <math>\{1,2,\ldots,n\}</math>. |
Revision as of 11:09, 17 April 2008
A group 
is a set of elements together with an operation 
(the dot is frequently supressed, so 
is written instead of 
) 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 
((Equivalently, a group is a monoids with inverses.)
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 or symmetries 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 in some texts 
(since it has 
elements) and in others 
(since it preserves a regular -gon). Another example of a group is the symmetric group 
of all permutations of 
.
See Also
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