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 ℵ 0 ), boundedly endo-rigid abelian group G of cardinality λ 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 ℵ 0 for which there is a co-Hopfian abelian group of size λ. 1. Introduction 183 2. Preliminaries 187 3. The ZFC construction of boundedly rigid mixed groups 191 4. Co-Hopfian and boundedly endo-rigid abelian groups 219 Acknowledgement 230 References 231