Monatshefte für Mathematik

, Volume 176, Issue 4, pp 497–502 | Cite as

On \(p\)-nilpotency of hyperfinite groups

  • A. Ballester-Bolinches
  • S. Camp-Mora
  • F. Spagnuolo
Article
  • 186 Downloads

Abstract

Let \(p\) be a prime. We say that class \(\fancyscript{X}\) of hyperfinite \(p\)-groups determines\(p\)-nilpotency locally if every finite group \(G\) with a Sylow \(p\)-subgroup \(P\) in \(\fancyscript{X}\) is \(p\)-nilpotent if and only if \({{\mathrm{N}}}_{G}(P)\) is \(p\)-nilpotent. The results of this paper improve a recent result of Kurdachenko and Otal and show that if a hyperfinite group \(G\) has a pronormal Sylow \(p\)-subgroup in \(\fancyscript{X}\), then \(G\) is \(p\)-nilpotent if and only if \({{\mathrm{N}}}_G(P)\) is \(p\)-nilpotent provided that \(\fancyscript{X}\) is closed under taking subgroups and epimorphic images. If \(\fancyscript{X}\) is not closed under taking epimorphic images, we have to impose local \(p\)-solubility to \(G\). In this case, the hypothesis of pronormality can be removed.

Keywords

Locally finite group Hyperfinite group \(p\)-Nilpotency 

Mathematics Subject Classification (2010)

20E15 20F19 20F22 

References

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Copyright information

© Springer-Verlag Wien 2014

Authors and Affiliations

  • A. Ballester-Bolinches
    • 1
  • S. Camp-Mora
    • 1
    • 2
  • F. Spagnuolo
    • 1
  1. 1.Departament d’ÀlgebraUniversitat de ValènciaValenciaSpain
  2. 2.Departament de Matemàtica AplicadaUniversitat Politècnica de ValènciaValenciaSpain

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