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Difference between revisions of "Semidirect product"

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In group theory, a '''semidirect product''' describes a particular way in which a group can be put together from two subgroups, one of which is [[normal subgroup|normal]].
 
In group theory, a '''semidirect product''' describes a particular way in which a group can be put together from two subgroups, one of which is [[normal subgroup|normal]].
  
Let ''G'' be a group, ''N'' a [[normal subgroup]] of ''G'' (i.e., ''N'' ◁ ''G'') and ''H'' a [[subgroup]] of ''G''. ''G'' is a '''semidirect product''' of ''N'' and ''H'' if there exists a [[group homomorphism|homomorphism]] ''G'' → ''H'' which is the identity on ''H'' and whose [[kernel (algebra)|kernel]] is ''N''. This is equivalent to say that:
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Let ''G'' be a group, ''N'' a [[normal subgroup]] of ''G'' (i.e., ''N'' ◁ ''G'') and ''H'' a [[subgroup]] of ''G''. ''G'' is a '''semidirect product''' of ''N'' and ''H'' if there exists a [[group homomorphism|homomorphism]] ''G'' → ''H'' which is the identity on ''H'' and whose [[Group homomorphism|kernel]] is ''N''. This is equivalent to say that:
 
* ''G'' = ''NH'' and ''N'' ∩ ''H'' = {1} (where "1" is identity element of ''G'' )
 
* ''G'' = ''NH'' and ''N'' ∩ ''H'' = {1} (where "1" is identity element of ''G'' )
 
* ''G'' = ''HN'' and ''N'' ∩ ''H'' = {1}
 
* ''G'' = ''HN'' and ''N'' ∩ ''H'' = {1}

Revision as of 18:31, 21 December 2008

Produit semi-direct (Fr). Semidirektes Produkt (Ge). Producto semidirecto (Sp). Полупрямое произведение (Ru). Prodotto semidiretto (It). 準直積 (Ja).


In group theory, a semidirect product describes a particular way in which a group can be put together from two subgroups, one of which is normal.

Let G be a group, N a normal subgroup of G (i.e., NG) and H a subgroup of G. G is a semidirect product of N and H if there exists a homomorphism GH which is the identity on H and whose kernel is N. This is equivalent to say that:

  • G = NH and NH = {1} (where "1" is identity element of G )
  • G = HN and NH = {1}
  • Every element of G can be written as a unique product of an element of N and an element of H
  • Every element of G can be written as a unique product of an element of H and an element of N

One also says that "G splits over N".