On problems by Baer and Kulikov using \(V=L\)
The author deals with two appealing problems of Abelian group theory. The first, motivated by work of R.~Baer, desires to determine, for a given torsion-free (Abelian) group \(G\), the class \(\mathcal{TC}(G)\) consisting of all torsion (Abelian) groups \(T\) such that \(\text{Ext}(G,T)=0\). Baer asked whether \(G\) has to be free if \(\mathcal{TC}(G)\) consists of all torsion groups. P.~Griffith proved in 1967 that this is indeed the case. The author mainly deals with the question whether \(\mathcal{TC}(G)=\mathcal{TC}(C)\) for some completely decomposable groups \(C\). Based on substantial results due to the author and S.~Wallutis, the following result is obtained. Corollary~2.9 (\(V=L\)) Assume that for all torsion-free groups \(H\) of size less than or equal to \(\aleph_1\) there exists a completely decomposable group \(C_H\) such that \(\mathcal{TC}(H)=\mathcal{TC}(C_H)\). Then for every torsion-free group \(G\) there is a completely decomposable group \(C\) such that \(\mathcal{TC}(G)=\mathcal{TC}(C)\). It is not known, but the author conjectures, that under the assumption \(V=L\), it is true in all cases that \(\mathcal{TC}(G)=\mathcal{TC}(C)\) for some completely decomposable group \(C\). There is evidence that the set theoretic hypothesis \(V=L\) is needed. The second problem is due to Kulikov. Given a torsion group \(T\) and a cardinal \(\lambda\), does there exist a torsion-free group \(G\) with the property that \(\text{Ext}(G,T)=0\) and whenever \(\text{Ext}(H,T)=0\) for some torsion-free group \(H\) with \(|H|<\lambda\) it follows that \(H\) is embedded in \(G\). Such a group \(G\) is called `\(\lambda\)-universal for \(T\)'. There is a neat general result for finite \(\lambda\). Theorem~3.3 Let \(T\) be a torsion group and \(n\in\mathbb{N}\). Then there is an \(n\)-universal group \(G\) for \(T\) if and only if the number of non-trivial bounded \(p\)-components of \(T\) is finite. If so, \(G\) is completely decomposable. For infinite \(\lambda\) the author quotes evidence that set-theoretic hypotheses are needed and proves the following result. Theorem~3.9 (\(V=L\)). Let \(T\) be a torsion group with a finite number of non-trivial bounded \(p\)-components and \(\lambda\) a cardinal. Then there is a \(\lambda\)-universal group \(G\) for \(T\) which is completely decomposable. The author conjectures that the last theorem holds for arbitrary \(T\). The paper is a pleasure to read.
scientific article; zbMATH DE number 1839956