On the Existence of Uncountable Hopfian and co-Hopfian Abelian Groups

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 ℵ 0 < λ < λ ℵ 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 extensions of the universe.

Co-Hopfian and boundedly endo-rigid mixed groups

“…By the same method one can prove the following variation of Theorem 4.8. Combining the previous results with the mentioned theorem of Paolini-Shelah [21], we observe the following:…”

“…Essentially, we give complete characterization of the pairs (K, λ) by relating our work with the recent works of Paolini and Shelah, see [20], [21] and [22]. To this end, first we recall the following folklore problem:…”

“…So, in the light of Theorem 1.3, the remaining part is 2 ℵ0 < λ < λ ℵ0 . Very recently, and among other things, Paolini and Shelah [21] proved that there is no co-Hopfian group of size λ for such a λ. As an application, in Section 4, we completely determine cardinals λ > 2 ℵ0 for which there is a co-Hopfian group of size λ.…”