Use with the (group-theory) tag. A group is cyclic if it can be generated by a single element. That is to say, every element in a cyclic group can be written as some specified element to a power.

A group $G$ is *cyclic* if it can be generated by a single element $a$. This means that any element of a cyclic group has the form $a^n$ for some integer $n$. Notationally, we often write that $G$ is isomorphic to $\langle a \rangle$. Since

$$a^na^m=a^{n+m}=a^{m+n}=a^ma^n\,,$$

cyclic groups must be abelian. Note though that the generator is not necessarily unique: for example the cyclic group $\mathbf{Z}/7\mathbf{Z}$, consisting of the elements $\{0,1,\dotsc,6\}$ and equipped with the operation of addition modulo $7$, can be generated by *any* of its non-identity elements.

Cyclic groups are completely classified. Up to isomorphism, $\mathbf{Z}$ equipped with addition is the only infinite cyclic group. Every finite cyclic group is isomorphic to a group of the form $\mathbf{Z}/n\mathbf{Z}$, a quotient of the integers under addition modulo $n$.

Cyclic groups are incredibly useful in describing the structure of finite abelian groups. By the classification theorem of finite abelian groups, every finite abelian group is isomorphic to a direct sum of cyclic groups, each having order a power of a prime.