Abstract
Abstract Let 𝐺 be a group. A subgroup 𝐻 of 𝐺 is called permutable if H K = K H HK=KH for all subgroups 𝐾 of 𝐺. Permutability is not in general a transitive relation, and 𝐺 is said to be a PT \mathrm{PT} -group (or to have the PT \mathrm{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 \mathrm{PT} -property. We prove that such a group is a PT \mathrm{PT} -group and so are all its subgroups.
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