Infinite groups with many permutable subgroups

Abstract

A subgroup $H$ of a group $G$ is said to be \textit{permutable in$G$}, if $HK = KH$ for every subgroup $K$ of $G$. A result due toStonehewer asserts that every permutable subgroup is ascendantalthough the converse is false. In this paper we study someinfinite 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 therelation \textit{to be a permutable subgroup} is transitive($PT$--groups).