# Difference between revisions of "Cyclic group"

From Encyclopedia of Mathematics

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− | A group with a single generator. All cyclic groups are Abelian. Every finite group of prime order is cyclic. For every finite number | + | {{TEX|done}} |

+ | 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 | ||

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

## Latest revision as of 10:03, 12 April 2014

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://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