On groups with many subgroups satisfying a transitive normality relation

Abstract

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