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 defining ingredient::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 defining ingredient::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