1.92: Semidirect product
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 ".