# Surjective homomorphism of groups

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## Definition

Suppose and are groups. A set map is termed a **surjective homomorphism of groups** from to if it satisfies the following:

- is a homomorphism of groups from to
**and**is surjective as a set map. - is a homomorphism of groups from to
**and**it is an epimorphism in the category of groups. - is a homomorphism of groups from to and it descends to an isomorphism of groups from the quotient group to where is the kernel of .

### 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.