# 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.

## 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 ?
i) If [mathematical expression not reproducible] is a finite field, then [mathematical expression not reproducible] is a cyclic group of order m and is generated by Frobenius automorphism [[phi].
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.
12] = 1), a cyclic group of order 12, 0((G [union] I)) = 60.
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.
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.
Breakthrough bleeding was reported by more women in the continuous treatment group (13%) than in the cyclic group (7%) (P = .
In particular, Jacques Wolfmann studied within this framework the notion of cyclic code (code which is invariant by the action of the cyclic group of permutations: particular case of affine groups).
The 68 women (mean age 28) who completed the study in the cyclic group applied a new patch weekly for 3 consecutive weeks, and then skipped a week to allow for bleeding.

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