Group monomorphism

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

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 $SL_m(\mathbb{R})$.