On soluble groups whose subnormal subgroups are inert

A subgroup H of a group G is called inert if‎, ‎for each g\in G‎, ‎the index of H\cap 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.

Keywords:

‎commensurable‎ , ‎strongly inert‎ , ‎finitely generated‎ , ‎HNN-extension