Exact factorizations of some classes of finite groups

Let G be a finite group and let A,B be subgroups of G. If G=AB={ab | a in A y b in B} then we say that G es factorizable and the pair (A,B) is called a factorization of G. We are going to say that a factorization (A,B) is exact if A n B={1}. Let G be a finite group and let us denote by F_2 (G) the factorization number of G and by f_2 (G) the exact factorization number of G; that is, the number of different factorizations and exact factorizations of G, respectively. The main purpose of this work is to calculate the exact factorization number for a large family of finite groups, those who are called metacyclic and metabelian groups.