A group is called cyclic if it is generated by a single element. If we are writing the group operation multiplicatively this means that G={an : n in Z}. (If we write it additively, as for some abelian groups, then this means G={na: n in Z}).

An example of an infinite cyclic group is (Z,+) (generated by 1). A finite cyclic group is the group of nth complex roots of unity (generated by e2pii/n).

Two cyclic groups with the same order are isomorphic.

A standard notation for the cyclic group of order n is Cn.

Cyc"lic (s?k"l?k ∨ s?"kl?k), Cyc"lic*al (s?k"l?-kal), a. [Cf. F. cycluque, Gr., fr. See Cycle.]

Of or pertaining to a cycle or circle; moving in cycles; as, cyclical time.

Coleridge.

Cyclic chorus, the chorus which performed the songs and dances of the dithyrambic odes at Athens, dancing round the altar of Bacchus in a circle. -- Cyclic poets, certain epic poets who followed Homer, and wrote merely on the Trojan war and its heroes; -- so called because keeping within the circle of a singe subject. Also, any series or coterie of poets writing on one subject.

Milman.

 

© Webster 1913.

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