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Archiv der Mathematik

, Volume 109, Issue 4, pp 305–310 | Cite as

Normalisers of residuals of finite groups

  • A. Ballester-BolinchesEmail author
  • S. F. Kamornikov
  • H. Meng
Article
  • 97 Downloads

Abstract

Let \(\mathfrak {F}\) be a subgroup-closed saturated formation of finite groups containing all finite nilpotent groups, and let M be a subgroup of a finite group G normalising the \(\mathfrak {F}\)-residual of every non-subnormal subgroup of G. We show that M normalises the \(\mathfrak {F}\)-residual of every subgroup of G. This answers a question posed by Gong and Isaacs (Arch Math 108:1–7, 2017) when \(\mathfrak {F}\) is the formation of all finite supersoluble groups.

Keywords

Finite group Formation Residual Subnormality 

Mathematics Subject Classification

20D10 20D35 

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References

  1. 1.
    A.  Ballester-Bolinches and L.M. Ezquerro, Classes of Finite Groups, Mathematics and its Applications, 584, Springer, New York, 2006.zbMATHGoogle Scholar
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    A. Ballester-Bolinches and M. C. Pedraza-Aguilera, On minimal subgroups of finite groups, Acta Math. Hungar. 73 (1996), 335–342.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    K. Doerk and T. Hawkes, Finite Soluble Groups, De Gruyter Expositions in Mathematics, 4, Walter de Gruyter, Berlin, New York, 1992.CrossRefzbMATHGoogle Scholar
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    L. Gong and I. M. Isaacs, Normalizers of nilpotent residuals, Arch. Math. 108 (2017), 1–7.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • A. Ballester-Bolinches
    • 1
    Email author
  • S. F. Kamornikov
    • 2
  • H. Meng
    • 1
  1. 1.Departament de MatemàtiquesUniversitat de ValènciaBurjassotSpain
  2. 2.Department of MathematicsFrancisk Skorina Gomel State UniversityGomelBelarus

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