# Surjective homomorphism of groups

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## 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 $\varphi$.