Homomorphically Stable Abelian Groups

A group is said to be homomorphically stable with respect to another group if the union of the homomorphic images of the first group in the second group is a subgroup of the second group. A group is said to be homomorphically stable if it is homomorphically stable with respect to every group. It is shown that a group is homomorphically stable if it is homomorphically stable with respect to its double direct sum. In particular, given any group, the direct sum and the direct product of infinitely many copies of this group are homomorphically stable; all endocyclic groups are homomorphically stable as well. Necessary and sufficient conditions for the homomorphic stability of a fully transitive torsion-free group are found. It is proved that a group is homomorphically stable if and only if so is its reduced part, and a split group is homomorphically stable if and only if so is its torsion-free part. It is shown that every group is homomorphically stable with respect to every periodic group.