A subgroup $H$ of a group $G$ is called $f$-subnormal in $G$, if there is a finite sequence $H=H_0\le H_1\le\cdots\le H_k=G$ such that the predecessor is normal in the following term whenever the index is infinite. It follows from results of Lennox and Stonehewer that finitely generated groups of the title are finite-by-nilpotent. Two results for the general case as examples: $G$ is finite-by-solvable, every subgroup of $G/D(G)$ is subnormal and $D(G)$ is finite-by-nilpotent, where $D(G)$ is generated by all nilpotent residuals of finitely generated subgroups. -- Further, the authors consider groups in which every subgroup is a subgroup of finite index of a subnormal subgroup.
[H.Heineken (Würzburg)]
