Ulderico Dardano - Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università di Napoli “Federico II”, Via Cintia - Monte S. Angelo, I-۸۰۱۲۶ Napoli, Italy
Fausto De Mari - Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università di Napoli “Federico II”, Via Cintia - Monte S. Angelo, I-۸۰۱۲۶ Napoli, Italy

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)‎.

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