TY - JOUR
T1 - On the existence of uncountable Hopfian and co-Hopfian abelian groups
AU - Paolini, Gianluca
AU - Shelah, Saharon
N1 - Publisher Copyright:
© 2023, The Author(s).
PY - 2023/11
Y1 - 2023/11
N2 - 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 Torp(G), and that in particular there are no infinite reduced abelian p-groups of size < p. Secondly, we prove that if 2ℵ0<λ<λℵ0 , 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.
AB - 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 Torp(G), and that in particular there are no infinite reduced abelian p-groups of size < p. Secondly, we prove that if 2ℵ0<λ<λℵ0 , 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.
UR - http://www.scopus.com/inward/record.url?scp=85180431606&partnerID=8YFLogxK
U2 - 10.1007/s11856-023-2534-4
DO - 10.1007/s11856-023-2534-4
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AN - SCOPUS:85180431606
SN - 0021-2172
VL - 257
SP - 533
EP - 560
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
IS - 2
ER -