Butler groups that are unions of subgroups with countable typesets (Q689645)

Recall that a torsionfree group \(B\) is called a \(B_ 1\)-group if \(\text{Bext} (B,T)=0\) for any torsion group \(T\) and it is called a \(B_ 2\)-group if there is a continuous well-ordered ascending chain of pure subgroups \(B=\bigcup_{\alpha<\tau} B_ \alpha\) with rank one factors such that, for each \(\alpha<\tau\), \(B_{\alpha+1}= B_ \alpha+G_ \alpha\) holds for some finite rank Butler group \(G_ \alpha\). It is well known that \(B_ 1\)-groups with countable typesets are \(B_ 2\)-groups. Among other results the authors prove that a continuous well-ordered ascending union of pure subgroups with countable typesets is a \(B_ 2\)- group, provided it is a \(B_ 1\)-group. This yields under (CH) that \(B_ 1\)-pure subgroups of completely decomposable groups are \(B_ 2\)-groups.