# 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 $n$ there is one and, up to isomorphism, only one cyclic group of order $n$; there is also one infinite cyclic group, which is isomorphic to the additive group $\mathbf Z$ of integers. A finite cyclic group $G$ of order $n$ is isomorphic to the additive group of the ring of residues $\mathbf Z(n)$ modulo $n$ (and also to the group $\mathbf C(n)$ of (complex) $n$-th roots of unity). Every element $a$ of order $n$ can be taken as a generator of this group. Then

$$G=\{1=a^0=a^n,a,\ldots,a^{n-1}\}.$$

How to Cite This Entry:
Cyclic group. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Cyclic_group&oldid=31609
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article