On groups with many subgroups satisfying a transitive normality relation

Abstract

A group $G$ is said to be a $T$-group if normality in $G$ is a transitive relation. Clearly, as a simple group has the property $T$, it follows that $T$ is not subgroup closed. A group $G$ is called a $\bar T$-group if all its subgroups are $T$-groups. In this note the structure of groups all of whose (proper) subgroups either are nilpotent or satisfy the property $\bar T$ will be investigated.