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Surjective homomorphism of groups

4 bytes removed, 23:38, 29 June 2013
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Suppose <math>G</math> and <math>H</math> are [[group]]s. A set map <math>\varphialpha:G \to H</math> is termed a '''surjective homomorphism of groups''' from <math>G</math> to <math>H</math> if it satisfies the following:
# <math>\varphialpha</math> is a [[defining ingredient::homomorphism of groups]] from <math>G</math> to <math>H</math> '''and''' <math>\varphi</math> is surjective as a set map.# <math>\varphialpha</math> is a [[homomorphism of groups]] from <math>G</math> to <math>H</math> '''and''' it is an [[cattheory:epimorphism|epimorphism]] in the [[defining ingredient::category of groups]].# <math>\varphialpha</math> is a [[homomorphism of groups]] from <math>G</math> to <math>H</math> and it descends to an [[isomorphism of groups]] from the [[quotient group]] <math>G/K</math> to <math>H</math> where <math>K</math> is the [[kernel]] of <math>\varphi</math>.
===Equivalence of definitions===
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