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On the Lattice of all \({{\mathcal {H}}}\)-Permutable Subgroups of a Finite Group

  • Bin Hu
  • Jianhong HuangEmail author
  • Alexander N. Skiba
Original Paper
  • 3 Downloads

Abstract

Let G be a finite group and \(\sigma =\{\sigma _{i} | i\in I\}\) is a partition of all primes \(\mathbb {P}\). A set \({{{\mathcal {H}}}}\) of subgroups of G is called a complete Hall \(\sigma \)-set of G if every member \(\ne 1\) of \({{{\mathcal {H}}}}\) is a Hall \(\sigma _{i}\)-subgroup of G for some \(\sigma _{i} \in \sigma \) and \({\mathcal {H}}\) contains exactly one Hall \(\sigma _{i}\)-subgroup of G for every i such that \(\sigma _{i}\in \sigma (G)\); a \(\sigma \)-basis of G if \({{{\mathcal {H}}}}\) is a complete Hall \(\sigma \)-set of G such that each of the subgroups \(A, B\in {{{\mathcal {H}}}}\) permutable. For any set \({{\mathcal {H}}}\) of subgroups of G, we write \({{{\mathcal {P}}}}({{\mathcal {H}}})\) to denote the set of all subgroups A of G such that A permutes with all subgroups \(B\in {{{\mathcal {H}}}}\), that is, \(AB=BA\); \({{{\mathcal {L}}}}_{{{{\mathcal {H}}}}{\mathfrak {S}}}(G)\) to denote the set of all subgroups A of G with \(A\in {{{\mathcal {P}}}}({{\mathcal {H}}})\) and soluble \(A^{G}/A_{G}\). We prove that the set \({{{\mathcal {P}}}} ({{\mathcal {H}}})\) and the set \({{{\mathcal {L}}}}_{{{{\mathcal {H}}}}{\mathfrak {S}}}(G)\) form sublattices of the lattice \({{{\mathcal {L}}}}(G)\) of all subgroups of G for each \(\sigma \)-basis \({{\mathcal {H}}}\) of G. We prove also that if all members of \({{\mathcal {H}}}\) are nilpotent, then the lattice \({{{\mathcal {L}}}}_{{{\mathcal {H}}}{\mathfrak {S}}}(G)\) is modular if and only if each of the two subgroups \(A, B \in {{{\mathcal {L}}}}_{{{{\mathcal {H}}}}{\mathfrak {S}}}(G)\) are permutable.

Keywords

Finite group \(\sigma \)-Basis Permutable subgroups Subgroup lattice Modular lattice 

Mathematics Subject Classification

20D10 20D30 20E15 

Notes

Acknowledgements

The authors are deeply grateful to the helpful suggestions of the referee.

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.

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Copyright information

© Iranian Mathematical Society 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsJiangsu Normal UniversityXuzhouPeople’s Republic of China
  2. 2.Department of Mathematics and Technologies of ProgrammingFrancisk Skorina Gomel State UniversityGomelBelarus

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