Group monomorphism - Imported Knowlpedia Content

A group monomorphism is a group homomorphism $\varphi\colon G\to H$ that is injective as an injective function.

Equivalent characterizations

Equivalently, $\varphi$ is a monomorphism if and only if its kernel is trivial: $\ker(\varphi)=\{e_G\}$ . In that case $\varphi$ identifies $G$ with the image $\varphi(G)$ , which is a subgroup of $H$ .

Examples

If $H\le G$ , the inclusion $H\hookrightarrow G$ is a group monomorphism.
The map $\mathbb{Z}\to\mathbb{Z}$ given by $n\mapsto 2n$ (additive groups) is a group monomorphism.
The determinant map $\det\colon GL_m(\mathbb{R})\to\mathbb{R}^\times$ is not a monomorphism for $m\ge 2$ because it has nontrivial kernel $SL_m(\mathbb{R})$ .