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)

@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}
}