# 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:

 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