A result on \(B_ 1\)-groups (Q1915405)

This paper repairs a gap in the proof of a theorem of \textit{D. Arnold} [in Boll. Unione Mat. Ital., VI. Ser., A 5, 175-184 (1986; Zbl 0601.20050)] as well as the proof of Bican-Salce that a countable \(B_1\)-group is finitely Butler. Later results relying on these two are thereby validated as well. Recall that a torsion-free Abelian group \(G\) is a \(B_1\)-group if \(\text{Bext}(G,T)=0\) for all torsion-groups \(T\); and \(G\) is finitely Butler if any pure finite rank subgroup is a Butler group. A subgroup \(K\) of a torsion-free Abelian group \(G\) is called generalized regular if \(G/K\) is torsion and for any rank-one pure subgroup \(X\) of \(G\), the \(p\)-component of \(X/(X\cap K)\) is zero for almost all primes \(p\). Theorem. Let \(H\) be an arbitrary \(B_1\)-group and \(B\) a generalized regular subgroup of \(H\). Then for any finite rank pure subgroup \(L\) of \(H\), the torsion group \((L+B)/B\) has at most finitely many nonzero \(p\)-components.