Groups whose derived subgroup is not supplemented by any proper subgroup

Abstract

In this paper, we introduce two new classes of groups that are described as weakly nilpotent and weakly solvable groups. A group $G$ is weakly nilpotent if its derived subgroup does not have a supplement except $G$ and a group $G$ is weakly solvable if its derived subgroup does not have a normal supplement except $G$. 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 $F$ is a free group of rank $n$ such that every normal subgroup of $F$ has rank $n$, then $F$ is weakly solvable.