Infinite groups with many permutable subgroups

Abstract

A subgroup $H$ of a group $G$ is said to be permutable in $G$, if $HK=KH$ for every subgroup $K$ of $G$. A result due to Stonehewer asserts that every permutable subgroup is ascendant although the converse is false. In this paper we study some infinite groups whose ascendant subgroups are permutable ($AP$-groups). We show that the structure of radical hyperfinite $AP$-groups behave as that of finite soluble groups in which the relation to be a permutable subgroup is transitive ($PT$-groups).