Groups whose derived subgroup is not supplemented by any proper subgroup - Narain, Shiv; Kumar, Sunil; Mittal, Gaurav; Kumar, Sandeep (Armenian Journal of Mathematics, 2022)
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.