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Difference between revisions of "Cyclic group"

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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027510/c0275101.png" /> there is one and, up to isomorphism, only one cyclic group of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027510/c0275102.png" />; there is also one infinite cyclic group, which is isomorphic to the additive group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027510/c0275103.png" /> of integers. A finite cyclic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027510/c0275104.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027510/c0275105.png" /> is isomorphic to the additive group of the ring of residues <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027510/c0275106.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027510/c0275107.png" /> (and also to the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027510/c0275108.png" /> of (complex) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027510/c0275109.png" />-th roots of unity). Every element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027510/c02751010.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027510/c02751011.png" /> can be taken as a generator of this group. Then
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027510/c02751012.png" /></td> </tr></table>
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$$G=\{1=a^0=a^n,a,\ldots,a^{n-1}\}.$$

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