Let $G$ and $H$ be groups, and let $\varphi: G \to H$ be a [group homomorphism](/page/Group%20Homomorphism). Let $g \in G$, and suppose $G=\langle g\rangle$, where $\langle g\rangle=\{g^n : n \in \mathbb{Z}\}$ is the cyclic subgroup generated by $g$, $\mathbb{Z}$ denotes the set of integers, and $\mathbb{N}=\{1,2,3,\dots\}$ denotes the set of positive integers. Then

\begin{align*} \operatorname{im}\varphi=\langle \varphi(g)\rangle. \end{align*}

In particular, every homomorphic image of a [cyclic group](/page/Cyclic%20Group) is cyclic.