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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 $n$, then every element of $G$ can be expressed as $x^{k}$ with $k\in\{0,\dots,n1\}$, 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 $C_{\infty}$, 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:

Every cyclic group is abelian.

Every subgroup of a cyclic group is cyclic.

Every quotient of a cyclic group is cyclic.

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

Every finite subgroup of the multiplicative group of a field is cyclic.
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Attached Articles
proof that all cyclic groups of the same order are isomorphic to each other by Wkbj79
proof that all cyclic groups are abelian by Wkbj79
proof that all subgroups of a cyclic group are cyclic by Wkbj79
Proof: The orbit of any element of a group is a subgroup by drini
$C_{mn}\cong C_m\times C_n$ when $m, n$ are relatively prime by yesitis
automorphism group of a cyclic group by rm50
subgroups of finite cyclic group by pahio