# Cyclic group

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

How to Cite This Entry:
Cyclic group. O.A. Ivanova (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Cyclic_group&oldid=13750
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098