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 $k\in\{0,\dots,n-1\}$ , 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 $C_{\infty}$ , 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 $\phi(n)$ generators, where $\phi$ 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	2013-03-22 12:23:27
Last modified on	2013-03-22 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