Image of a group homomorphism - Imported Knowlpedia Content

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 $H$ .

Remarks

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 $\mathrm{im}(\iota)=H$ .