Semidirect product

(Difference between revisions)
 Revision as of 08:42, 20 November 2017 (view source)m (Tidied translations.)← Older edit Latest revision as of 11:32, 15 December 2017 (view source)m (Tidied translations.) Line 1: Line 1: - Produit semi-direct (''Fr''). Semidirektes Produkt (''Ge''). Полупрямое произведение (''Ru''). Prodotto semidiretto (''It''). 準直積 (''Ja''). Producto semidirecto (''Sp''). + Produit semi-direct (''Fr''). Semidirektes Produkt (''Ge''). Prodotto semidiretto (''It''). 準直積 (''Ja''). Полупрямое произведение (''Ru''). Producto semidirecto (''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'.