# Cyclic Group

Also found in: Wikipedia.

## cyclic group

[′sīk·lik ‚grüp] (mathematics)

A group that has an element

*a*such that any element in the group can be expressed in the form*a*, where^{n}*n*is an integer.McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

The following article is from

*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.## Cyclic Group

in mathematics, a group for which all elements are powers of one element. The set of *n*th roots of unity is an example of a finite cyclic group. The set of integers forms an infinite cyclic group under addition (since the group operation in this case is addition, multiples are considered instead of powers). All finite cyclic groups with the same number of elements are isomorphic, as are all infinite cyclic groups. Any subgroup and any quotient group of a cyclic group are cyclic groups.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.