# Cyclic group

From Encyclopedia of Mathematics

A group with a single generator. All cyclic groups are Abelian. Every finite group of prime order is cyclic. For every finite number there is one and, up to isomorphism, only one cyclic group of order ; there is also one infinite cyclic group, which is isomorphic to the additive group of integers. A finite cyclic group of order is isomorphic to the additive group of the ring of residues modulo (and also to the group of (complex) -th roots of unity). Every element of order can be taken as a generator of this group. Then

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Cyclic group.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Cyclic_group&oldid=13750

This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article