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

Authors: Gábor Braun and Lutz Strüngmann
Journal: Proc. Amer. Math. Soc. 143 (2015), 3331-3341
MSC (2010): Primary 20K30; Secondary 20K15
Published electronically: April 23, 2015
MathSciNet review: 3348775
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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:
\begin{inparaenum}[(a)]\item both Hopfian and co-Hopfian, \item Hopfian but not co-Hopfian, \item co-Hopfian but not Hopfian. \end{inparaenum}
All three types of groups of size $ 2^{\aleph _0}$ exist in ZFC.

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Gábor Braun
Affiliation: Fakultät für Mathematik, Universität DuisburgEssen, Campus Essen, 45117 Essen, Germany
Address at time of publication: H. Milton Stewart School of Industrial and Systems Engineering, Georgia Insitute of Technology, 755 Ferst Drive, NW Atlanta, Georgia 30332

Lutz Strüngmann
Affiliation: Faculty for Computer Sciences, Mannheim University of Applied Sciences, 68163 Mannheim, Germany

Keywords: Martin's axiom, Hopfian groups, co-Hopfian groups
Received by editor(s): February 1, 2012
Received by editor(s) in revised form: May 3, 2013, September 17, 2013, and November 11, 2013
Published electronically: April 23, 2015
Additional Notes: The first author’s research was partially supported by the Hungarian Scientific Research Fund, Grant No. NK 81203 and project STR 627/1-6 of the German Research Foundation DFG
Communicated by: Birge Huisgen-Zimmermann
Article copyright: © Copyright 2015 American Mathematical Society