cyclic group
A group is said to be cyclic if it is generated by a single element.
Suppose $G$ is a cyclic group^{} generated by $x\in G$. Then every element of $G$ is equal to ${x}^{k}$ for some $k\in \mathbb{Z}$. If $G$ is infinite, then these ${x}^{k}$ are all distinct, and $G$ is isomorphic^{} to the group $\mathbb{Z}$. If $G$ has finite order (http://planetmath.org/OrderGroup) $n$, then every element of $G$ can be expressed as ${x}^{k}$ with $k\in \{0,\mathrm{\dots},n1\}$, and $G$ is isomorphic to the quotient group^{} $\mathbb{Z}/n\mathbb{Z}$.
Note that the isomorphisms mentioned in the previous paragraph imply that all cyclic groups of the same order are isomorphic to one another. The infinite cyclic group is sometimes written ${C}_{\mathrm{\infty}}$, and the finite cyclic group of order $n$ is sometimes written ${C}_{n}$. However, when the cyclic groups are written additively, they are commonly represented by $\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}$.
While a cyclic group can, by definition, be generated by a single element, there are often a number of different elements that can be used as the generator^{}: an infinite cyclic group has $2$ generators, and a finite cyclic group of order $n$ has $\varphi (n)$ generators, where $\varphi $ is the Euler totient function.
Some basic facts about cyclic groups:

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Every cyclic group is abelian^{}.

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Every subgroup^{} of a cyclic group is cyclic.

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Every quotient of a cyclic group is cyclic.

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Every group of prime order is cyclic. (This follows immediately from Lagrange’s Theorem.)

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Every finite subgroup of the multiplicative group^{} of a field is cyclic.
Title  cyclic group 
Canonical name  CyclicGroup 
Date of creation  20130322 12:23:27 
Last modified on  20130322 12:23:27 
Owner  yark (2760) 
Last modified by  yark (2760) 
Numerical id  21 
Author  yark (2760) 
Entry type  Definition 
Classification  msc 20A05 
Related topic  GeneralizedCyclicGroup 
Related topic  PolycyclicGroup 
Related topic  VirtuallyCyclicGroup 
Related topic  CyclicRing3 
Defines  cyclic 
Defines  cyclic subgroup 
Defines  infinite cyclic 
Defines  infinite cyclic group 
Defines  infinite cyclic subgroup 