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

Abstract

AbstractWe 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 $${2^{{\aleph _0}}} < \lambda < {\lambda ^{{\aleph _0}}}$$ 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 extension of the universe.