Cyclic group/Z mod n/Example

Let ${}G$ be a cyclic group with a generator ${}g$. We consider the corresponding group homomorphism

$\varphi \colon \mathbb {Z} \longrightarrow G,n\longmapsto g^{n},$

in the sense of fact. Since we have a generator, this mapping is surjective. The kernel of this mapping is determined the order of ${}g$, which we denote by ${}k$ (or it is ${}0$, in case the order is ${}\infty$). Due to fact, there exists a canonical isomorphism

${\tilde {\varphi }}\colon \mathbb {Z} /(k)\longrightarrow G.$

In particular, there exists, up to isomorphism, for every ${}k$, exactly one cyclic group, namely ${}\mathbb {Z} /(k)$.