On soluble groups whose subnormal subgroups are inert

Abstract

A subgroup H of a group G is called inert if‎, ‎for each $gin G$‎, ‎the index of $Hcap H^g$ in $H$ is finite‎. ‎We give a classification of soluble-by-finite groups $G$ in which subnormal subgroups are inert in the cases where $G$ has no nontrivial torsion normal subgroups or $G$ is finitely generated‎.