Infinite groups with many permutable subgroups

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).