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

Let 𝐺 be a group. A subgroup 𝐻 of 𝐺 is called permutable if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>H</m:mi> <m:mo>⁢</m:mo> <m:mi>X</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>⁢</m:mo> <m:mi>H</m:mi> </m:mrow> </m:mrow> </m:math> HX=XH for all subgroups 𝑋 of 𝐺. Permutability is not in general a transitive relation, and 𝐺 is called a <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>PT</m:mi> </m:math> 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>PT</m:mi> </m:math> PT is not inherited by subgroups, and 𝐺 is called a <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mover accent="true"> <m:mi>PT</m:mi> <m:mo>̄</m:mo> </m:mover> </m:math> PT̄ -group if all its subgroups have the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>PT</m:mi> </m:math> PT -property. We prove that if 𝐺 is a soluble group of infinite rank whose proper subgroups of infinite rank have the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>PT</m:mi> </m:math> PT -property, then 𝐺 is a <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mover accent="true"> <m:mi>PT</m:mi> <m:mo>̄</m:mo> </m:mover> </m:math> PT̄ -group.