On cosmall Abelian groups. (Q2466932)

In the present paper the authors investigate what happens with Abelian groups with dual properties of some well known homological characterizations of (self-)small Abelian groups (modules). More precisely, an Abelian group \(G\) is called `cosmall' if \(\Hom(\prod_{i\in I}A_i,G)\) and \(\prod_{i\in I}\Hom(A_i,G)\) are naturally isomorphic for all families \((A_i\mid i\in I)\). If we ask that the mentioned isomorphism is valid for families \((A_i\mid i\in I)\) with \(A_i\cong G\) for all \(i\), then \(G\) is called `self-cosmall'. The authors investigate basic properties of these groups, and they prove that all these groups are torsion-free (Theorem 2.2 and Proposition 3.2), products of cosmall groups are cosmall (Proposition 2.3). Moreover in Section 2 it is proved that some very important classes of torsion-free groups are not cosmall, and these results suggest that there are no non-trivial cosmall groups and non-trivial self-cosmall groups. This interesting result is proved in the last section of the paper under the assumptions that there exists a strongly compact cardinal for cosmall groups (Theorem 3.3), respectively there exists a proper class of strongly compact cardinals (Corollary 3.5). As the authors mention in the end of the paper, the questions for the existence of non-trivial cosmall groups or of non-trivial self-cosmall groups in ZFC or in ZFC + V=L are still open.