# 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. arXiv: math/0012172 DOI: 10.1515/form.2003.028 MR: 1978332 -
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} }