On torsion and mixed minimal Abelian groups.
As a way of investigating the concept of minimality in a general categorical setting, the authors call an Abelian group \(G\) minimal if it is isomorphic to every subgroup of finite index in \(G\). In 1998 Robinson and Timm explored this concept in non-Abelian group theory under the title of ``hc-groups''. In the present work, the authors obtain a characterization of torsion minimal groups in terms of their Ulm invariants, and characterizations of certain mixed torsion-free rank 1 minimal groups. In the torsion case, Goldsmith and hÓgáin show that a torsion group is minimal if and only if all of its primary components are minimal. Furthermore, if \(G\) is a reduced \(p\)-group, then \(G\) is minimal if and only if there exists an ordinal \(\lambda\leq\omega\) such that \(f_n(G)\) is infinite for all \(n<\lambda\) and, if \(\lambda<\omega\), \(f_n(G)=0\) for all \(n\geq\lambda\). The authors list several properties of minimal mixed groups and obtain several characterizations. One of them is that for certain torsion free rank 1 mixed groups \(G\), \(G\) is minimal if and only if \(t(G)\) and \(G/t(G)\) are minimal.
scientific article; zbMATH DE number 2012083
scientific article