Groups whose derived subgroup is not supplemented by any proper subgroup

In this paper, we introduce two new classes of groups that are described as weakly nilpotent and weakly solvable groups. A group GG is weakly nilpotent if its derived subgroup does not have a supplement except GG and a group GG is weakly solvable if its derived subgroup does not have a normal supplement except GG. We present some examples and counter-examples for these groups and characterize a finitely generated weakly nilpotent group. Moreover, we characterize the nilpotent and solvable groups in terms of weakly nilpotent and weakly solvable groups. Finally, we prove that if FF is a free group of rank nn such that every normal subgroup of FF has rank nn, then FF is weakly solvable.