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

Document Type : Research Paper

Authors

Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università di Napoli “Federico II”, Via Cintia - Monte S. Angelo, I-80126 Napoli, Italy

Abstract

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)‎.

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