Groups whose Proper Subgroups of Infinite Rank Have a Transitive Normality Relation

Abstract

A group G is called a T-group if all its subnormal subgroups are normal, and G is a \({\bar{T}}\) -group if every subgroup of G has the property T. It is proved here that if G is a locally soluble group whose proper subgroups of infinite rank have the T-property, then either G is a \({\bar{T}}\) -group or it has finite rank.