Cyclic Group

(redirected from Infinite cyclic)

cyclic group

[′sīk·lik ‚grüp]
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 ?
In the fibered case, we work on an associated infinite cyclic covering of M.
If F is non-separating, we work on the infinite cyclic covering space of M defined by a cohomology class dual to F.
be the infinite cyclic covering of M, given by the cohomology class dual to a choice of fibering.
There are lifts of the resulting spun normal surfaces in each direction, which bounds a closed normal surface in the infinite cyclic covering [?
0], we see that the infinite cyclic subgroup [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.
Take a hyperbolic element [gamma] [member of] [GAMMA], that is [absolute value of tr([gamma])] > 2,then the centralizer of [gamma] in [GAMMA] is infinite cyclic and [gamma] is conjugate in PSL(2, R) to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with N([gamma]) > l.