# Cyclic Group

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## 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 It might be outdated or ideologically biased.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.

References in periodicals archive ?
Li and associates found 228 cases of VTE in the noncyclic group and 297 cases in the cyclic group, with an incidence rate of 1.54 (95% confidence interval, 1.34-1.74) per 1,000 person-years for noncyclic users and 0.83 (95% CI, 0.74-0.93) per 1,000 person-years for cyclic users (crude hazard ratio, 1.84; 95% CI, 1.53-2.21).
Then its unit group [F.sup.x] is a cyclic group of order n -1.
The cyclic group of order [p.sup.n] is denoted by [mathematical expression not reproducible].
In one-legged stance with eyes open, there was a significant increase in postural displacement in the mid-luteal phase in the cyclic group, but no differences were detected between phases in the non-cyclic group (Friden et al.
Let [pi] = Cay([Z.sub.n], [OMEGA]) be the Cayley graph on the cyclic group [Z.sub.n] (n [greater than or equal to] 4), where [OMEGA] = {t [member of] [Z.sub.n] | t is odd} is the inverse-closed subset of [Z.sub.n] -{0}.
Given a generator P of an additive cyclic group G with order q and given (aP, bP) for unknown a, b [member of] [Z.sup.*.sub.q], one computes abP.
There exist two non-isomorphic groups of order 34: the cyclic group of order 34 and the dihedral group of order 34.
The numbers 0 through 10 form the cyclic group or commutative ring upon which summation (mod 11) and multiplication (mod 11) are defined.
A smaller proportion of women in the continuous regimen group than in the cyclic group reported backache at six months (39% vs.
Since [PSI] is invertible, it generates a cyclic group ([PSI]} acting on J(P), but the orbit structure is not immediately apparent.
The Group Calculator allows the user to select a group from one of several families: cyclic group of order n, dihedral group [D.sub.n] of order 2n, the group [Z.sup.*.sub.n] of units modulo n, the abelian group [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and the affine group Aff([Z.sub.n]) = {ax + b | a [member of] [Z.sup.*.sub.n] and b [member of] [Z.sub.n]} under composition.

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