If the group G=AB is the product of two abelian subgroups A and B, then G is metabelian by a well-known result of Itô , so that the commutator subgroup G' of G is abelian. In the following we are concerned with the following condition:
There exists a normal subgroup
which is contained in A or B.
Recently, Holt and Howlett in  have given an example of a countably infinite p-group G = AB, which is the product of two elementary abelian subgroups A and B with Core(A) = Core (B) = 1, so that in this group (*) does not hold. Also, Sysak in  gives an example of a product G = AB of two free abelian subgroups A and B with Core(A)=Core(B)=l.