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.