Semidirect product

From Online Dictionary of Crystallography

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<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'').
<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'').

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'.