Group epimorphism

A group epimorphism is a group homomorphism $\varphi\colon G\to H$ that is surjective as a surjective function .

Equivalently, $\varphi$ is an epimorphism if and only if its image equals all of $H$, i.e. $\varphi(G)=H$. Many natural epimorphisms arise as quotient maps: if $N\lhd G$ then the canonical projection $G\to$ quotient group $G/N$ is surjective.

Examples:

• The reduction map $\mathbb{Z}\to\mathbb{Z}/n\mathbb{Z}$ is an epimorphism of additive groups.
• The projection $G\times H\to G$ is a group epimorphism.
• The sign map $\mathrm{sgn}\colon S_n\to\{\pm 1\}$ is a group epimorphism for $n\ge 2$.