Groups whose proper subgroups of infinite rank have a permutability transitive relation

Abstract

Abstract Let 𝐺 be a group. A subgroup 𝐻 of 𝐺 is called permutable if H ⁢ X = X ⁢ H HX=XH for all subgroups 𝑋 of 𝐺. Permutability is not in general a transitive relation, and 𝐺 is called a PT \mathrm{PT} -group if, whenever 𝐾 is a permutable subgroup of 𝐺 and 𝐻 is a permutable subgroup of 𝐾, we always have that 𝐻 is permutable in 𝐺. The property PT \mathrm{PT} is not inherited by subgroups, and 𝐺 is called a PT ̄ \overline{\mathrm{PT}} -group if all its subgroups have the PT \mathrm{PT} -property. We prove that if 𝐺 is a soluble group of infinite rank whose proper subgroups of infinite rank have the PT \mathrm{PT} -property, then 𝐺 is a PT ̄ \overline{\mathrm{PT}} -group.