Non-existence of universal members in classes of Abelian groups. (Q2711629)

The author considers the question of the existence of universal members in classes of Abelian groups. This problem has also been studied in previous papers by \textit{M. Kojman} and the author [Isr. J. Math. 92, No. 1-3, 113-124 (1995; Zbl 0840.20057)] and by the author [Algebra Log. Appl. 9, 229-286 (1997; Zbl 0936.20044)].NEWLINENEWLINE For a class \(\mathfrak K\) of Abelian groups and a cardinal \(\lambda\) we denote by \({\mathfrak K}_\lambda\) the class of groups in \(\mathfrak K\) of cardinality \(\lambda\); a group \(U\) of \({\mathfrak K}_\lambda\) is universal in \({\mathfrak K}_\lambda\) if any other group in \({\mathfrak K}_\lambda\) can be embedded into \(U\). The author proves the non-existence for certain classes \(\mathfrak K\) and for certain cardinals \(\lambda\). More precisely, it is shown that there is no universal element in \({\mathfrak K}_\lambda\) for the class \({\mathfrak K}={\mathfrak K}^{rs(p)}\) of all reduced separable \(p\)-groups and for all \(\lambda\) with \(\beth_\omega\leq\mu^+<\lambda=cf(\lambda)<\mu^{\aleph_0}\) (for some \(\mu\)), and also for the class \({\mathfrak K}={\mathfrak K}^{rtf}\) of all reduced torsion-free Abelian groups and for all \(\lambda\) with \(\aleph_0<\lambda<2^{\aleph_0}\) or \(2^{\aleph_0}<\mu^+<\lambda=cf(\lambda)<\mu^{\aleph_0}\) or \(\beth_\omega\leq\mu^+<\lambda=cf(\lambda)<\mu^{\aleph_0}\); in the latter case there is not even a reduced torsion-free group which is universal for all \(\aleph_1\)-free groups of cardinality \(\lambda\).