Difference between revisions of "Semidirect product"
From Online Dictionary of Crystallography
BrianMcMahon (talk | contribs) m (Style edits to align with printed edition) |
BrianMcMahon (talk | contribs) m (Tidied translations.) |
||
| Line 1: | Line 1: | ||
| − | <font color="blue">Produit semi-direct</font> (''Fr''). <font color="red">Semidirektes Produkt</font> (''Ge''). <font color=" | + | <font color="blue">Produit semi-direct</font> (''Fr''). <font color="red">Semidirektes Produkt</font> (''Ge''). <font color="brown">Полупрямое произведение</font> (''Ru''). <font color="black">Prodotto semidiretto</font> (''It''). <font color="purple">準直積</font> (''Ja''). <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]]. | ||
Revision as of 08:42, 20 November 2017
Produit semi-direct (Fr). Semidirektes Produkt (Ge). Полупрямое произведение (Ru). Prodotto semidiretto (It). 準直積 (Ja). 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. 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'.