Abstract
We deal with the problem of existence of uncountable co-Hopfian abelian groups and (absolute) Hopfian abelian groups. Firstly, we prove that there are no co-Hopfian reduced abelian groups G of size < p with infinite Torp(G), and that in particular there are no infinite reduced abelian p-groups of size < p. Secondly, we prove that if \({2^{{\aleph _0}}} < \lambda < {\lambda ^{{\aleph _0}}}\), and G is abelian of size λ, then G is not co-Hopfian. Finally, we prove that for every cardinal λ there is a torsion-free abelian group G of size λ which is absolutely Hopfian, i.e., G is Hopfian and G remains Hopfian in every forcing extension of the universe.
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This paper is dedicated to Moshe Jarden, in honor of his many contributions to field arithmetic
No. 1214 on Shelah’s publication list. Research of both authors partially supported by NSF grant no: DMS 1833363. Research of first author partially supported by project PRIN 2017 “Mathematical Logic: models, sets, computability”, prot. 2017NWTM8R. Research of second author partially supported by Israel Science Foundation (ISF) grant no: 1838/19.
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Paolini, G., Shelah, S. On the existence of uncountable Hopfian and co-Hopfian abelian groups. Isr. J. Math. 257, 533–560 (2023). https://doi.org/10.1007/s11856-023-2534-4
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DOI: https://doi.org/10.1007/s11856-023-2534-4