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\cap 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)‎.

Keywords:

‎transitivity‎ , ‎core-finite‎ , ‎normal-by-finite‎ , ‎close to normal