Having learned how isomorphisms preserve structure, we can classify the simplest family of groups. The focus keyword classification of cyclic groups means that cyclic groups are determined, up to isomorphism, entirely by their order. A finite cyclic group of order $n$ is structurally the same as $\mathbb{Z}_n$, and an infinite cyclic group is structurally the same as $\mathbb{Z}$. This result is powerful because it converts an abstract generator into a familiar additive model. Students often think two cyclic groups differ because one is written multiplicatively and another additively. Classification shows that notation is secondary; the order of the generator controls the whole group.

Every finite cyclic group of order $n$ is isomorphic to $(\mathbb{Z}_n,+_n)$.