Image of a group homomorphism

Let $\varphi\colon G\to H$ be a group homomorphism . The image of $\varphi$ is the subset $\mathrm{im}(\varphi)=\varphi(G)=\{\varphi(g): g\in G\}\subseteq H.$ The image is always a subgroup of $H$.

The map $\varphi$ is a group epimorphism if and only if $\mathrm{im}(\varphi)=H$. Together with the kernel, the image appears in the first isomorphism theorem , which identifies $G/\ker(\varphi)$ with $\mathrm{im}(\varphi)$ as groups.

Examples:

• For $\varphi\colon\mathbb{Z}\to\mathbb{Z}$ defined by $\varphi(n)=2n$, one has $\mathrm{im}(\varphi)=2\mathbb{Z}$.
• For $\mathrm{sgn}\colon S_n\to\{\pm1\}$, the image is all of $\{\pm1\}$ (for $n\ge 2$).
• If $\iota\colon H\hookrightarrow G$ is the inclusion of a subgroup, then $\mathrm{im}(\iota)=H$.