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 $n$ 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 | + | A group with a single generator. All cyclic groups are [[Abelian group|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 |
$$G=\{1=a^0=a^n,a,\ldots,a^{n-1}\}.$$ | $$G=\{1=a^0=a^n,a,\ldots,a^{n-1}\}.$$ | ||
Latest revision as of 06:43, 21 March 2024
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=31609
Cyclic group. Encyclopedia of Mathematics. URL: http://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