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Chemistry LibreTexts

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 GG 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".