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Difference between revisions of "Semidirect product"

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<font color="blue">Produit semi-direct</font> (''Fr''). <font color="red">Semidirektes Produkt</font> (''Ge''). <font color="green">Producto semidirecto</font> (''Sp''). <font color="brown">Полупрямое произведение</font> (''Ru''). <font color="black">Prodotto semidiretto</font> (''It''). <font color="purple">準直積</font> (''Ja'').  
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<font color="blue">Produit semi-direct</font> (''Fr''). <font color="red">Semidirektes Produkt</font> (''Ge''). <font color="black">Prodotto semidiretto</font> (''It''). <font color="purple">準直積</font> (''Ja''). <font color="brown">Полупрямое произведение</font> (''Ru''). <font color="green">Producto semidirecto</font> (''Sp'').
  
  
 
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'' &#x25C1; ''G'') and ''H'' a [[subgroup]] of ''G''. ''G'' is a '''semidirect product''' of ''N'' and ''H'' if there exists a [[group homomorphism|homomorphism]] ''G'' &rarr; ''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'' &#x25C1; ''G'') and ''H'' a [[subgroup]] of ''G''. ''G'' is a '''semidirect product''' of ''N'' and ''H'' if there exists a [[group homomorphism|homomorphism]] ''G'' &rarr; ''H'' which is the identity on ''H'' and whose [[Group homomorphism|kernel]] is ''N''. This is equivalent to saying that:
* ''G'' = ''NH'' and ''N'' &cap; ''H'' = {1} (where "1" is identity element of ''G'' )
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* ''G'' = ''NH'' and ''N'' &cap; ''H'' = {1} (where '1' is the identity element of ''G'').
* ''G'' = ''HN'' and ''N'' &cap; ''H'' = {1}
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* ''G'' = ''HN'' and ''N'' &cap; ''H'' = {1}.
* Every element of ''G'' can be written as a unique product of an element of ''N'' and an element of ''H''
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* 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''
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* 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''".
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One also says that `''G'' ''splits'' over ''N'''.
  
 
[[Category:Fundamental crystallography]]
 
[[Category:Fundamental crystallography]]

Latest revision as of 11:32, 15 December 2017

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


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 saying that:

  • G = NH and NH = {1} (where '1' is the 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'.