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

TL;DR

This paper investigates soluble groups where subnormal subgroups of infinite rank are closely related to normal subgroups, showing that under certain conditions, all subnormal subgroups are commensurable with a normal subgroup.

Contribution

It establishes conditions under which subnormal subgroups of infinite rank are commensurable with normal subgroups in periodic soluble groups.

Findings

If the Hirsch-Plotkin radical has infinite rank, all subnormal subgroups are commensurable with a normal subgroup.

In nilpotent-by-abelian groups with infinite rank, the same property holds.

The results extend understanding of subgroup structure in infinite rank soluble groups.

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 the intersection of H and 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).