Let $(G,\cdot)$ be a group with identity element $e_G$. Suppose that $g \in G$, that $G=\langle g \rangle$, and that $\operatorname{ord}(g)=\infty$. Define the map
paragraph
admin
\begin{align*}
\varphi: \mathbb{Z} \to G
\end{align*}
latex_env
admin
by $\varphi(n)=g^n$ for every $n \in \mathbb{Z}$, where $g^n$ denotes the integer power of $g$ in the group $G$. Then $\varphi$ is a group isomorphism from $(\mathbb{Z},+)$ onto $G$. In particular,
paragraph
admin
\begin{align*}
G \cong (\mathbb{Z},+).
\end{align*}