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 Tor(p)(G), and that in particular there are no infinite reduced abelian p-groups of size < lambda. Secondly, we prove that if 2(N0) < lambda < lambda(N0), and G is abelian of size lambda, then G is not co-Hopfian. Finally, we prove that for every cardinal. there is a torsion-free abelian group G of size. which is absolutely Hopfian, i. e., G is Hopfian and G remains Hopfian in every forcing extension of the universe.
On the existence of uncountable Hopfian and co-Hopfian abelian groups
Paolini, Gianluca
;
2023-01-01
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 Tor(p)(G), and that in particular there are no infinite reduced abelian p-groups of size < lambda. Secondly, we prove that if 2(N0) < lambda < lambda(N0), and G is abelian of size lambda, then G is not co-Hopfian. Finally, we prove that for every cardinal. there is a torsion-free abelian group G of size. which is absolutely Hopfian, i. e., G is Hopfian and G remains Hopfian in every forcing extension of the universe.
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