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Periodica Mathematica Hungarica

, Volume 69, Issue 1, pp 69–78 | Cite as

Fully inert subgroups of free Abelian groups

  • D. Dikranjan
  • L. SalceEmail author
  • P. Zanardo
Article

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 \(0 \rightarrow H \rightarrow G \rightarrow G/H \rightarrow 0\).

Keywords

Abelian groups Fully inert subgroups Free groups 

Mathematics Subject Classification

Primary: 20K27 Secondary: 20K20 20K15 

Notes

Acknowledgments

This research supported by “Progetti di Eccellenza 2011/12” of Fondazione CARIPARO.

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2014

Authors and Affiliations

  1. 1.Dipartimento di Matematica e InformaticaUniversità di UdineUdineItaly
  2. 2.Dipartimento di MatematicaPadovaItaly

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