### It is consistent with ZFC that $B_1$-groups are not $B_2$-groups

by Shelah and Struengmann. [ShSm:754]

Forum Math, 2003

A torsion-free abelian group B of arbitrary rank is called a
B_1-group if Bext^1(B,T)=0 for every torsion abelian group
T, where 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 ... subset B_alpha subset ... 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.

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