Finite Groups with Systems of Generalized Normal Subgroups

Let be a finite group and let be the lattice of all subnormal subgroups of . Let and be subgroups of and let be a sublattice in ; i.e., , for all . Then: is the -closure of in ; i.e., the intersection of all subgroups in which includes and is the -core of in , i.e., the subgroup in generated by all subgroups of belonging to . A subgroup is an - -subgroup in if either or and avoids each composition factor of between and ; i.e., . Using these notions, we give some new characterizations of soluble and supersoluble subgroups and generalize a few available results.