Groups whose all subgroups are ascendant or self-normalizing

This paper studies groups $G$ whose subgroups either are ascendant or self-normalizing. We characterize the structure of such $G$ in case they are locally finite. If $G$ is a hyperabelian group and has the property, we show that every subgroup of $G$ is in fact ascendant provided $G$ is locally nilpotent or non-periodic. We also restrict our study replacing ascendant subgroups by permutable subgroups, which of course are ascendant (\cite{SS}).}