CO-HOPFIAN AND BOUNDEDLY ENDO-RIGID MIXED ABELIAN GROUPS

For a given cardinal λ and a torsion abelian group K of cardinality less than λ, we present, under some mild conditions (for example, λ = λ^S0), boundedly endo-rigid abelian group G of cardinality 1 with tor(G) = K. Essentially, we give a complete characterization of such pairs (K, λ). Among other things, we use a twofold version of the black box. We present an application of the construction of boundedly endo-rigid abelian groups. Namely, we turn to the existence problem of co-Hopfian abelian groups of a given size, and present some new classes of them, mainly in the case of mixed abelian groups. In particular, we give useful criteria to detect when a boundedly endo-rigid abelian group is co-Hopfian and completely determine cardinals λ > 2^S0 for which there is a co-Hopfian abelian group of size λ.