Semidirect Product - Some Equivalent Definitions

Some Equivalent Definitions

Let G be a group with identity element e, N a normal subgroup of G (i.e., N ◁ G) and H a subgroup of G. The following statements are equivalent:

• G = NH and N ∩ H = {e}.
• G = HN and N ∩ H = {e}.
• 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.
• The natural embedding H → G, composed with the natural projection G → G / N, yields an isomorphism between H and the quotient group G / N.
• There exists a homomorphism G → H which is the identity on H and whose kernel is N.

If one (and therefore all) of these statements hold, we say that G is a semidirect product of N and H, written or that G splits over N; one also says that G is a semidirect product of H acting on N, or even a semidirect product of H and N. In order to avoid ambiguities, it is advisable to specify which of the two subgroups is normal.