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).

**Keywords**

**Main Subjects**

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**Receive Date:**24 January 2021**Revise Date:**06 April 2021**Accept Date:**12 April 2021**Published Online:**01 March 2022