Two Generalizations of co-Hopfian Abelian Groups

TL;DR

This paper introduces and thoroughly characterizes generalized co-Hopfian and relatively co-Hopfian abelian groups, providing classifications and structural descriptions for various subclasses including p-groups, torsion-free, and mixed groups.

Contribution

It offers the first comprehensive classification and structural analysis of generalized co-Hopfian and relatively co-Hopfian abelian groups, extending classical concepts.

Findings

Complete classification of generalized co-Hopfian p-groups.

Structural description of torsion-free generalized co-Hopfian groups.

Equivalence of super and hereditarily relatively co-Hopfian groups.

Abstract

By defining the classes of generalized co-Hopfian and relatively co-Hopfian groups, respectively, we consider two expanded versions of the generalized co-Bassian groups and of the classical co-Hopfian groups giving a close relationship with them. Concretely, we completely describe generalized co-Hopfian p-groups for some prime p obtaining that such a group is either divisible, or it splits into a direct sum of a special bounded group and a special co-Hopfian group. Furthermore, a comprehensive description of a torsion-free generalized co-Hopfian group is obtained. In addition, we fully characterize when a mixed splitting group and, in certain cases, when a genuinely mixed group are generalized co-Hopfian. Finally, complete characterizations of a super hereditarily generalized co-Hopfian group as well as of a hereditarily generalized co-Hopfian group are given, showing in the latter situation that it decomposes as the direct sum of three specific summands. Moreover, we totally classify relatively co-Hopfian p-groups proving the unexpected fact that they are exactly the co-Hopfian ones. About the torsion-free and mixed cases, we show in light of direct decompositions that in certain situations they are satisfactory classifiable -- e.g., the splitting mixed relatively co-Hopfian groups and the relatively co-Hopfian completely decomposable torsion-free groups. Finally, complete classifications of super and hereditarily relatively co-Hopfian groups are established in terms of ranks which rich us that these two classes curiously do coincide.