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

Braun, Gábor; Strüngmann, Lutz

doi:10.1090/proc/12413

The independence of the notions of Hopfian and co-Hopfian Abelian p-groups
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by Gábor Braun and Lutz Strüngmann PDF

Proc. Amer. Math. Soc. 143 (2015), 3331-3341 Request permission

Abstract:

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 $2^{\aleph _0}$ is independent of ZFC: (a) both Hopfian and co-Hopfian, (b) Hopfian but not co-Hopfian, (c) co-Hopfian but not Hopfian. All three types of groups of size $2^{\aleph _0}$ exist in ZFC.

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