# 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 ?
There exist finite groups that are semidirect products of cyclic groups and are not determined by their endomorphism semigroups in the class of all groups [10].
On the definability of a semidirect product of cyclic groups by its endomorphism semigroup.
k] - the cyclic group of order k) are determined by their endomorphism semigroups in the class of all groups.
Semidirect products of generalized quaternion groups by a cyclic group.
Among the topics are greatest common divisors, integer multiples and exponents, quotients of polynomial rings, divisibility and factorization in integral domains, subgroups of cyclic groups, cosets and Lagrange's theorem, the fundamental theorem of finite abelian groups, and check digits.
Popular choices for the group G in discrete logarithm cryptography are the cyclic groups [[].
Let G be a finite cyclic group, and [theta] [member of] G be a generator of G.
Let G = ([theta]) be a cyclic group of order n, and let a and b be elements 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].
Polyrotaxanes are polymers with cyclic groups threaded onto a polymer chain.
They focus mainly on cyclic groups, even though some of the results become trivial.
Mathematicians and scientists discuss such matters as rigid abelian groups and the probabilistic method, looking for indecomposable right bounded complexes, kernel modules of cotorsion pairs, upper cardinal bounds for absolute structures, subgroups of totally projective primary abelian groups and direct sums of cyclic groups, generic endomorphisms of homogeneous structures, special pairs and automorphisms of centerless groups, and some results on the algebraic entropy.

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