Difference between revisions of "Semidirect product"
From Online Dictionary of Crystallography
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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 [[homomorphism]] ''G'' → ''H'' which is the identity on ''H'' and whose [[kernel (algebra)|kernel]] is ''N''. This is equivalent to say that: | + | 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: |
* ''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 10:00, 29 May 2007
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 say that:
- G = NH and N ∩ H = {1} (where "1" is 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".