Two Generalizations of Hopfian Abelian Groups
Fuente:
arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913461403910144 |
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| author | Chekhlov, Andrey R. Danchev, Peter V. Goldsmith, Brendan Keef, Patrick W. |
| author_facet | Chekhlov, Andrey R. Danchev, Peter V. Goldsmith, Brendan Keef, Patrick W. |
| contents | This paper targets to generalize the notion of Hopfian groups in the commutative case by defining the so-called {\bf relatively Hopfian groups} and {\bf weakly Hopfian groups}, and establishing some their crucial properties and characterizations. Specifically, we prove that for a reduced Abelian $p$-group $G$ such that $p^ωG$ is Hopfian (in particular, is finite), the notions of relative Hopficity and ordinary Hopficity do coincide. We also show that if $G$ is a reduced Abelian $p$-group such that $p^ωG$ is bounded and $G/p^ωG$ is Hopfian, then $G$ is relatively Hopfian. This allows us to construct a reduced relatively Hopfian Abelian $p$-group $G$ with $p^ωG$ an infinite elementary group such that $G$ is {\bf not} Hopfian. In contrast, for reduced torsion-free groups, we establish that the relative and ordinary Hopficity are equivalent. Moreover, the mixed case is explored as well, showing that the structure of both relatively and weakly Hopfian groups can be quite complicated. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_01277 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Two Generalizations of Hopfian Abelian Groups Chekhlov, Andrey R. Danchev, Peter V. Goldsmith, Brendan Keef, Patrick W. Group Theory Commutative Algebra This paper targets to generalize the notion of Hopfian groups in the commutative case by defining the so-called {\bf relatively Hopfian groups} and {\bf weakly Hopfian groups}, and establishing some their crucial properties and characterizations. Specifically, we prove that for a reduced Abelian $p$-group $G$ such that $p^ωG$ is Hopfian (in particular, is finite), the notions of relative Hopficity and ordinary Hopficity do coincide. We also show that if $G$ is a reduced Abelian $p$-group such that $p^ωG$ is bounded and $G/p^ωG$ is Hopfian, then $G$ is relatively Hopfian. This allows us to construct a reduced relatively Hopfian Abelian $p$-group $G$ with $p^ωG$ an infinite elementary group such that $G$ is {\bf not} Hopfian. In contrast, for reduced torsion-free groups, we establish that the relative and ordinary Hopficity are equivalent. Moreover, the mixed case is explored as well, showing that the structure of both relatively and weakly Hopfian groups can be quite complicated. |
| title | Two Generalizations of Hopfian Abelian Groups |
| topic | Group Theory Commutative Algebra |
| url | https://arxiv.org/abs/2408.01277 |