Two Generalizations of Co-Hopfian Abelian Groups

By introducing the classes of generalized co-Hopfian groups and relatively co-Hopfian groups, we study two natural extensions of generalized co-Bassian groups and of classical co-Hopfian groups, thereby clarifying the close relationships between these notions. Specifically, we completely describe generalized co-Hopfian $ p $ -groups for a given prime $ p $ and prove that each such group is either divisible or splits as a direct sum of a special bounded group and a special co-Hopfian group. Furthermore, we obtain a comprehensive description of torsion-free generalized co-Hopfian groups and of mixed splitting groups. In addition, we characterize, in several cases, when a genuinely mixed group is generalized co-Hopfian. Finally, we provide complete characterizations of super hereditarily generalized co-Hopfian groups as well as of hereditarily generalized co-Hopfian groups. Moreover, we fully classify relatively co-Hopfian $ p $ -groups, establishing the unexpected fact that they coincide precisely with the co-Hopfian ones. For the torsion-free and mixed cases, we show, using direct decompositions, that in certain situations these groups admit satisfactory classifications, for instance, splitting mixed relatively co-Hopfian groups and relatively co-Hopfian completely decomposable torsion-free groups. Finally, complete classifications of super and hereditarily relatively co-Hopfian groups are obtained in terms of ranks, showing that these two classes coincide.