Groups with locally defined heights and products of \({\mathfrak R}^*\) groups (Q2738752)

Recall, that a subgroup \(B\) of a group \(G\) is said to be precobalanced if for any homomorphism \(\alpha\) from \(B\) into a rank one group \(R\) there is a finite rank torsionfree completely decomposable group \(H\) containing \(R\) as a pure subgroup such that \(\alpha\) extends to \(\beta\colon G\to H\). The class of all torsionfree groups such that each rank one pure subgroup is precobalanced is denoted by \({\mathfrak R}^*\).NEWLINENEWLINENEWLINEA vector group \(\prod_{\lambda\in I}R_\lambda\) is an \({\mathfrak R}^*\)-group if and only if (1) there is no infinite sequence of distinct labels \(\lambda_1,\lambda_2,\dots\) such that for all \(n\) \(\bigwedge_{k=1}^{n+1}{\mathbf t}(R_{\lambda_k})<\bigwedge_{k=1}^n{\mathbf t}(R_{\lambda_k})\) (here \({\mathbf t}(R)\) denotes the type of the rank one group \(R\)); (2) there is no infinite sequence of distinct labels \(\lambda_1,\lambda_2,\dots\) and distinct primes \(p_1,p_2,\dots\) such that for all \(n\) \(p_nR_{\lambda_n}\neq R_{\lambda_n}\) and \({\mathbf t}(R)\leq{\mathbf t}(R_{\lambda_n})\) where \(\mathbb{Z}\subseteq R\subseteq\mathbb{Q}\) is such that the \(p\)-height of 1 is 1 if \(p\in\{p_1,p_2,\dots\}\) and it is 0 otherwise (Theorem 1).NEWLINENEWLINENEWLINEA group \(\prod_{\lambda\in I}B_\lambda\) is an \({\mathfrak R}^*\)-group if and only if (1) each \(B_\lambda\) is an \({\mathfrak R}^*\)-group; (2) there is no infinite sequence of distinct labels \(\lambda_1,\lambda_2,\dots\) and types \(\tau_n\) from the typeset of \(B_{\lambda_n}\) such that for all \(n\) \(\bigwedge_{k=1}^{n+1}\tau_k<\bigwedge_{k=1}^n\tau_k\); (3) there is no infinite sequence of distinct labels \(\lambda_1,\lambda_2,\dots\), types \(\tau_n\) from the typeset of \(B_{\lambda_n}\), and distinct primes \(p_1,p_2,\dots\) such that \({\mathbf t}(R)\leq\tau_n\) where \(\tau_n\) has finite \(p_n\)-component for all \(n\), and \(\mathbb{Z}\subseteq R\subseteq\mathbb{Q}\) is such that the \(p\)-height of 1 is 1 if \(p\in\{p_1,p_2,\dots\}\) and it is 0 otherwise (Theorem 2).NEWLINENEWLINEFor the entire collection see [Zbl 0960.00043].