Surjective homomorphism of groups

Definition

Suppose $G$ and $H$ are groups. A set map $\alpha:G \to H$ is termed a surjective homomorphism of groups from $G$ to $H$ if it satisfies the following:

1. $\alpha$ is a homomorphism of groups from $G$ to $H$ and $\varphi$ is surjective as a set map.
2. $\alpha$ is a homomorphism of groups from $G$ to $H$ and it is an epimorphism in the category of groups.
3. $\alpha$ is a homomorphism of groups from $G$ to $H$ and it descends to an isomorphism of groups from the quotient group $G/K$ to $H$ where $K$ is the kernel of $\varphi$.

Equivalence of definitions

• Epimorphism iff surjective in the category of groups demonstrates the equivalence of (1) and (2).
• The equivalence of (1) and (3) follows from the first isomorphism theorem.

Related notions

• Surjective endomorphism