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''). | <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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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. | + | 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 saying that: |
| − | * ''G'' = ''NH'' and ''N'' ∩ ''H'' = {1} (where | + | * ''G'' = ''NH'' and ''N'' ∩ ''H'' = {1} (where '1' is the identity element of ''G''). |
| − | * ''G'' = ''HN'' and ''N'' ∩ ''H'' = {1} | + | * ''G'' = ''HN'' and ''N'' ∩ ''H'' = {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 ''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'' | + | * 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 | + | One also says that `''G'' ''splits'' over ''N'''. |
[[Category:Fundamental crystallography]] | [[Category:Fundamental crystallography]] | ||
Revision as of 10:32, 17 May 2017
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. N ◁ G) and H a subgroup of G. G is a semidirect product of N and H if there exists a homomorphism G → H which is the identity on H and whose kernel is N. This is equivalent to saying that:
- G = NH and N ∩ H = {1} (where '1' is the identity element of G).
- G = HN and N ∩ H = {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'.