## Quick Reference

A set *G* on which there is defined a dyadic operation ° (mapping *G* × *G* into *G*) that satisfies the following properties:*e* in *G* with the property that *x* ° *e* = *e* ° *x* = *x* for all *x* in *G*; *e* is called the **identity** of the group;**inverses** exist in *G*, i.e. for each *x* in *G* there is an inverse, denoted by *x*^{-1}, with the property that *x* ° *x*^{-1} = *x*^{-1} ° *x* = *e* These are the group axioms.Certain kinds of groups are of particular interest. If the dyadic operation ° is commutative, the group is said to be a commutative group or an abelian group (named for the Norwegian mathematician Niels Abel).

*e* in *G* with the property that *x* ° *e* = *e* ° *x* = *x* for all *x* in *G*; *e* is called the **identity** of the group;

**inverses** exist in *G*, i.e. for each *x* in *G* there is an inverse, denoted by *x*^{-1}, with the property that *x* ° *x*^{-1} = *x*^{-1} ° *x* = *e* These are the group axioms.

If there is only a finite number of elements *n* in the group, the group is said to be **finite**; *n* is then the **order** of the group. Finite groups can be represented or depicted by means of a Cayley table.

If the group has a generator then it is said to be **cyclic**; a cyclic group must be abelian.

The group is a very important algebraic structure that underlies many other algebraic structures such as rings and fields. There are direct applications of groups in the study of symmetry, in the study of transformations and in particular permutations, and also in error detecting and error correcting as well as in the design of fast adders.

Groups were originally introduced for solving an algebraic problem. By group theory it can be shown that algorithmic methods of a particular kind cannot exist for finding the roots of a general polynomial of degree greater than four. See also semigroup.

**From:**
group
in
A Dictionary of Computing »

*Subjects:*
Computing.

## Related content in Oxford Index

##### Reference entries

Users without a subscription are not able to see the full content. Please, subscribe or login to access all content.