Difference between revisions of "Surjective homomorphism of groups"

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(Created page with "==Definition== Suppose <math>G</math> and <math>H</math> are groups. A set map <math>\varphi:G \to H</math> is termed a '''surjective homomorphism of groups''' from <math...")
 
 
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==Definition==
 
==Definition==
  
Suppose <math>G</math> and <math>H</math> are [[group]]s. A set map <math>\varphi:G \to H</math> is termed a '''surjective homomorphism of groups''' from <math>G</math> to <math>H</math> if it satisfies the following:
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Suppose <math>G</math> and <math>H</math> are [[group]]s. A set map <math>\alpha: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>\varphi</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.
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# <math>\alpha</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>\varphi</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]].
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# <math>\alpha</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>\varphi</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>.
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# <math>\alpha</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===
 
===Equivalence of definitions===

Latest revision as of 23:38, 29 June 2013

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.

Equivalence of definitions

Related notions