Groups in which each subgroup is commensurable with a normal subgroup

A group G is a CN-group if for each subgroup H of G there exists a normal subgroup N of G such that the index |HN:(H∩N)| is finite. The class of CN-groups contains properly the classes of core-finite groups and that of groups in which each subgroup has finite index in a normal subgroup. In the present paper it is shown that a CN-group whose periodic images are locally finite is finite-by-abelian-by-finite. Such groups are then described into some details by considering automorphisms of abelian groups. Finally, it is shown that if G is a locally graded group with the property that the above index is bounded independently of H, then G is finite-by-abelian-by-finite.