# Sh:568

• Göbel, R., & Shelah, S. (2001). Some nasty reflexive groups. Math. Z., 237(3), 547–559.
• Abstract:
In Almost Free Modules, Set-theoretic Methods, p. 455, Problem 12, Eklof and Mekler raised the question about the existence of dual abelian groups G which are not isomorphic to {\mathbb Z} \oplus G. Recall that G is a dual group if G \cong D^* for some group D with D^*={\rm Hom}(D,{\mathbb Z}). The existence of such groups is not obvious because dual groups are subgroups of cartesian products {\mathbb Z}^D and therefore have very many homomorphisms into {\mathbb Z}. If \pi is such a homomorphism arising from a projection of the cartesian product, then D^*\cong{\rm ker}\pi \oplus {\mathbb Z}. In all “classical cases” of groups D of infinite rank it turns out that D^*\cong{\rm ker}\pi. Is this always the case? Also note that reflexive groups G in the sense of H. Bass are dual groups because by definition the evaluation map \sigma:G\longrightarrow G^{**} is an isomorphism, hence G is the dual of G^*. Assuming the diamond axiom for \aleph_1 we will construct a reflexive torsion-free abelian group of cardinality \aleph_1 which is not isomorphic to {\mathbb Z}\oplus G. The result is formulated for modules over countable principal ideal domains which are not field.
• published version (13p)
Bib entry
@article{Sh:568,
author = {G{\"o}bel, R{\"u}diger and Shelah, Saharon},
title = {{Some nasty reflexive groups}},
journal = {Math. Z.},
fjournal = {Mathematische Zeitschrift},
volume = {237},
number = {3},
year = {2001},
pages = {547--559},
issn = {0025-5874},
mrnumber = {1845337},
mrclass = {20K20 (03E35 20K30)},
doi = {10.1007/PL00004879},
note = {\href{https://arxiv.org/abs/math/0003164}{arXiv: math/0003164}},
arxiv_number = {math/0003164}
}