# Cyclic Group

(redirected from Finite cyclic group)

## 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 an, where 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 It might be outdated or ideologically biased.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 nth 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.
References in periodicals archive ?
If G is a finite cyclic group, then #[A.sup.G] = #[A.sub.G] holds.
Among the topics are partition functions and box-spline, the calculus of operator functions, toric Sasaki-Einstein geometry, automorphic representations, rigidity of polyhedral surfaces, number theory techniques in the theory of Lie groups and differential geometry, the flabby glass group of a finite cyclic group, Green's formula in Hall algebras and cluster algebras, zeta functions in combinatorics and number theory, and soliton hierarchies constructed from involutions.
Let G be a finite cyclic group, and [theta] [member of] G be a generator of G.
The discrete logarithm problem (DPL) in a finite cyclic group G, with a generator [theta] and an element g, is to find the integer a; 0 [less than or equal to] a [less than or equal to] [absolute value of G]-1, such that g = [[theta].sup.a] holds.
Clearly [Z.sub.2][S.sub.n] contains a finite cyclic group of order [2.sub.n]p.
They illustrate their method with free groups, triangular groups, and finite cyclic groups, for which they obtain optimal time hypercontractive L2 &gt; Lq inequalities with respect to the Markov process given by the world length and with q an even integer.

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