On Strongly Invariant Subgroups of Abelian Groups

It is shown that every homogeneous separable torsion-free group is strongly invariant simple (i.e., has no nontrivial strongly invariant subgroups) and, for a completely decomposable torsion-free group, every strongly invariant subgroup coincides with some direct summand of the group. The strongly invariant subgroups of torsion-free separable groups are described. In a torsion-free group of finite rank, every strongly inert subgroup is commensurable with some strongly invariant subgroup if and only if the group is free. The periodic groups, torsion-free groups, and split mixed groups in which every fully invariant subgroup is strongly invariant are described.