TOTALLY INERT SUBGROUPS OF ABELIAN GROUPS

A subgroup H of an abelian group G is totally inert if, for every nonzero endomorphism ϕ of G, H is commensurable with ϕ(H), that is, H∩ϕ(H) has finite index in H and in ϕ(H). In this paper totally inert subgroups of abelian groups are investigated. It is proved that, if a group is decomposable, then the only totally inert subgroups are the finite ones. So the investigation is addressed to torsion-free indecomposable groups. Totally inert subgroups of several indecomposable torsion-free groups are examined, both of finite and infinite rank. Characterizations are provided of torsion-free groups all of whose subgroups are totally inert, as well as of torsion-free groups of finite rank which are totally inert in themselves, which we have called self totally inert.