Abstract

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.

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