Group

Revision as of 09:21, 18 July 2006 by JBL (talk | contribs)

This article is a stub. Help us out by expanding it.


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:

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 $D_{2n}$ (since it has $2n$ elements). 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.