On the existence of uncountable Hopfian and co-Hopfian abelian groups

2021/07/23 by Paolini, Gianluca, Shelah, Saharon
#03E75 #20K30 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2107.11290

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 < \mathfrakp with infinite Tor_p(G), and that in particular there are no infinite reduced abelian p-groups of size < \mathfrakp. Secondly, we prove that if 2^ℵ₀ < λ< λ^ℵ₀, 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 extensions of the universe.

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