Difference between revisions of "Group"

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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 <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>.
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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>.
  
 
Related algebraic structures are [[ring]]s and [[field]]s.
 
Related algebraic structures are [[ring]]s and [[field]]s.

Revision as of 15:53, 16 October 2006

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A group $G$ is a set of elements together with an operation $\cdot:G\times G\to G$ (the dot is frequently supressed) satisfying the following conditions:


One can also thing of groups as 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 of collections of objects. For example, the rigid motions of $\mathbb{R}^2$ that fix a certain regular $n$-gon is a group, called the dihedral group and denoted in some texts $D_{2n}$ (since it has $2n$ elements) and in others $D_n$ (since it preserves a regular $n$-gon). Another example of a group is the symmetric group $S_n$ of all permutations of $\{1,2,\ldots,n\}$.

Related algebraic structures are rings and fields.