on groups in which subnormal subgroups of infinite rank are commensurable‎ ‎with some normal subgroup

We study soluble groups g in which each subnormal subgroup h with infinite rank is‎ ‎commensurable with a normal subgroup‎, ‎i.e‎. ‎there‎ ‎exists a normal subgroup n such that h∩n has finite index‎ ‎in both h and n‎. ‎we show that if such a g is periodic‎, ‎then‎ ‎all subnormal subgroups are commensurable with a normal subgroup‎, ‎provided either the hirsch-plotkin radical of g has infinite‎ ‎rank or g is nilpotent-by-abelian (and has infinite rank)‎.