Abstract
A subgroup \(H\) of an Abelian group \(G\) is called fully inert if \((\phi H + H)/H\) is finite for every \(\phi \in \mathrm{End}(G)\). Fully inert subgroups of free Abelian groups are characterized. It is proved that \(H\) is fully inert in the free group \(G\) if and only if it is commensurable with \(n G\) for some \(n \ge 0\), that is, \((H + nG)/H\) and \((H + nG)/nG\) are both finite. From this fact we derive a more structural characterization of fully inert subgroups \(H\) of free groups \(G\), in terms of the Ulm–Kaplansky invariants of \(G/H\) and the Hill–Megibben invariants of the exact sequence \(0 \rightarrow H \rightarrow G \rightarrow G/H \rightarrow 0\).
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Acknowledgments
This research supported by “Progetti di Eccellenza 2011/12” of Fondazione CARIPARO.
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This article dedicated to László Fuchs on the occasion of his 90th birthday.
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Dikranjan, D., Salce, L. & Zanardo, P. Fully inert subgroups of free Abelian groups. Period Math Hung 69, 69–78 (2014). https://doi.org/10.1007/s10998-014-0041-4
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DOI: https://doi.org/10.1007/s10998-014-0041-4