# Sh:754

• Shelah, S., & Strüngmann, L. H. (2003). It is consistent with ZFC that B_1-groups are not B_2. Forum Math., 15(4), 507–524.
• Abstract:
A torsion-free abelian group B of arbitrary rank is called a B_1-group if {\rm Bext}^1(B,T)=0 for every torsion abelian group T, where {\rm Bext}^1 denotes the group of equivalence classes of all balanced exact extensions of T by B. It is a long-standing problem whether or not the class of B_1-groups coincides with the class of B_2-groups. A torsion-free abelian group B is called a B_2-group if there exists a continuous well-ordered ascending chain of pure subgroups, 0=B_0 \subset B_1 \subset\cdots\subset B_\alpha\subset\cdots\subset B_\lambda=B= \bigcup\limits_{\alpha\in\lambda} B_\alpha such that B_{\alpha+1} =B_\alpha+G_\alpha for every \alpha\in\lambda for some finite rank Butler group G_\alpha. Both, B_1-groups and B_2-groups are natural generalizations of finite rank Butler groups to the infinite rank case and it is known that every B_2-group is a B_1-group. Moreover, assuming V=L it was proven that the two classes coincide. Here we demonstrate that it is undecidable in ZFC whether or not all B_1-groups are B_2-groups. Using Cohen forcing we prove that there is a model of ZFC in which there exists a B_1-group that is not a B_2-group.
• Version 2001-10-30_11 (19p) published version (18p)
Bib entry
@article{Sh:754,
author = {Shelah, Saharon and Str{\"u}ngmann, Lutz H.},
title = {{It is consistent with ZFC that $B_1$-groups are not $B_2$}},
journal = {Forum Math.},
fjournal = {Forum Mathematicum},
volume = {15},
number = {4},
year = {2003},
pages = {507--524},
issn = {0933-7741},
mrnumber = {1978332},
mrclass = {20K20 (03E35 20A15 20K15)},
doi = {10.1515/form.2003.028},
note = {\href{https://arxiv.org/abs/math/0012172}{arXiv: math/0012172}},
arxiv_number = {math/0012172}
}