Uncountable groups in which permutability is a transitive relation

Let G be a group. A subgroup H of G is called permutable if HK=KH for all subgroups K of G. Permutability is not in general a transitive relation, and G is said to be a PT-group (or to have the PT-property) if, whenever K is a permutable subgroup of G and H is a permutable subgroup of K, H is permutable in G. The aim of this paper is to investigate the behaviour of uncountable soluble groups of cardinality ℵ whose proper subgroups of cardinality ℵ have the PT-property. We prove that such a group is a PT-group and so are all its subgroups.