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.
Read more about this topic: Semidirect Product
Famous quotes containing the words equivalent and/or definitions:
“The notion that one will not survive a particular catastrophe is, in general terms, a comfort since it is equivalent to abolishing the catastrophe.”
—Iris Murdoch (b. 1919)
“The loosening, for some people, of rigid role definitions for men and women has shown that dads can be great at calming babies—if they take the time and make the effort to learn how. It’s that time and effort that not only teaches the dad how to calm the babies, but also turns him into a parent, just as the time and effort the mother puts into the babies turns her into a parent.”
—Pamela Patrick Novotny (20th century)