Title: Groups in which each subgroup is commensurable with a normal subgroup
Abstract: 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 both the wellknown classes of core-finite groups and of finite-by-abelian groups.In the present paper it is shown that a cn-group whose periodic images are locally finite is finite-by-abelian-by-finite.Then such groups are 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. 1