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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.