Semidirect product
From Online Dictionary of Crystallography
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. 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'.