# Changes

## Surjective homomorphism of groups

, 23:38, 29 June 2013
no edit summary
==Definition==
Suppose $G$ and $H$ are [[group]]s. A set map $\varphialpha:G \to H$ is termed a '''surjective homomorphism of groups''' from $G$ to $H$ if it satisfies the following:
# $\varphialpha$ is a [[defining ingredient::homomorphism of groups]] from $G$ to $H$ '''and''' $\varphi$ is surjective as a set map.# $\varphialpha$ is a [[homomorphism of groups]] from $G$ to $H$ '''and''' it is an [[cattheory:epimorphism|epimorphism]] in the [[defining ingredient::category of groups]].# $\varphialpha$ 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===