The independence of the notions of Hopfian and co-Hopfian Abelian p-groups

Hopfian and co-Hopfian Abelian groups have recently become of great interest in the study of algebraic and adjoint entropy. Here we prove that the existence of the following three types of infinite abelian p-groups of size less than 2₀ is independent of ZFC: (a)Â both Hopfian and co-Hopfian, (b)Â Hopfian but not co-Hopfian, (c)Â co-Hopfian but not Hopfian. All three types of groups of size 2₀ exist in ZFC.