Two Generalizations of co-Hopfian Abelian Groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866929683009896448 |
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| author | Chekhlov, Andrey R. Danchev, Peter V. Keef, Patrick W. |
| author_facet | Chekhlov, Andrey R. Danchev, Peter V. Keef, Patrick W. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_11452 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Two Generalizations of co-Hopfian Abelian Groups Chekhlov, Andrey R. Danchev, Peter V. Keef, Patrick W. Group Theory Rings and Algebras 20K10, 20K12, 20K20, 20K21, 20K30 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. |
| title | Two Generalizations of co-Hopfian Abelian Groups |
| topic | Group Theory Rings and Algebras 20K10, 20K12, 20K20, 20K21, 20K30 |
| url | https://arxiv.org/abs/2501.11452 |