Finite Factorizable Groups with Metabelian Schmidt Subgroups in the Factors

A Schmidt group is a finite nonnilpotent group all of whose proper subgroups are nilpotent. We study a finite group G=AB under the assumption that all Schmidt subgroups in A and in B have equal derived lengths. In this situation, in particular, it is proved that if A and B are subnormal in G and the indices of the subgroups A and B are coprime, then all Schmidt subgroups in G have equal derived lengths. In addition, a characterization of finite groups in which every metabelian subgroup is nilpotent is obtained. In particular, such a group is 2 -closed and the derived length of each of its Schmidt subgroups is equal to 3. It follows that every nonnilpotent group with trivial Frattini subgroup possesses a metabelian Schmidt subgroup.