Siberian Mathematical Journal

, Volume 60, Issue 4, pp 720–726 | Cite as

On Strongly Π-Permutable Subgroups of a Finite Group

  • B. Hu
  • J. HuangEmail author
  • A. N. Skiba


Let σ = {σi | iI} be some partition of the set of all primes ℙ,let ∅ ≠ Π ⊆ σ, and let G be a finite group. A set of subgroups of G is said to be a complete Hall Π-set of G if each member ≠ 1 of is a Hall σi-subgroup of G for some σi ∈ Π and has exactly one Hall σi-subgroup of G for every σi ∈ Π such that σiπ(G) ≠ ∅. A subgroup A of G is called (i) Π-permutable in G if G has a complete Hall Π-set such that AHx = HxA for all H and xG; (ii) σ-subnormal in G if there is a subgroup chain A = A0A1 ≤ ⋯ ≤ At = G such that either Ai−1Ai or Ai/(Ai−1)Ai is a σk-group for some k for all i = 1,…,t; and (iii) strongly Π-permutable if A is Π-permutable and σ-subnormal in G. We study the strongly Π-permutable subgroups of G. In particular, we give characterizations of these subgroups and prove that the set of all strongly Π-permutable subgroups of G forms a sublattice of the lattice of all subgroups of G.


finite group subgroup lattice σ-subnormal subgroup strongly Π-permutable subgroup 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Skiba A. N., “On some results in the theory of finite partially soluble groups,” Commun. Math. Stat., vol. 4, no. 2, 281–309 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Skiba A. N., “On σ-subnormal and σ-permutable subgroups of finite groups,” J. Algebra, vol. 436, 1–16 (2015).MathSciNetCrossRefGoogle Scholar
  3. 3.
    Skiba A. N., “Some characterizations of finite σ-soluble PσT-groups,” J. Algebra, vol. 495, 114–129 (2018).MathSciNetCrossRefGoogle Scholar
  4. 4.
    Guo W. and Skiba A. N., “On Π-quasinornial subgroups of finite groups,” Monatsh. Math., vol. 185, 443–453 (2018).MathSciNetCrossRefGoogle Scholar
  5. 5.
    Kegel O. H., “Sylow-Gruppen und Subnormalteiler endlicher Gruppen,” Math. Z., Bd 78, 205–221 (1962).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ballester-Bolinches A., Esteban-Romero R., and Asaad M., Products of Finite Groups, Walter de Gruyter, Berlin and New York (2010).CrossRefzbMATHGoogle Scholar
  7. 7.
    Ballester-Bolinches A. and Esteban-Romero R., “On finite soluble groups in which Sylow permutability is a transitive relation,” Acta Math. Hungar., vol. 101, 193–202 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Skiba A. N., “On finite groups for which the lattice of S-permutable subgroups is distributive,” Arch. Math., vol. 109, 9–17 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Doerk K. and Hawkes T., Finite Soluble Groups, Walter de Gruyter, Berlin and New York (1992).CrossRefzbMATHGoogle Scholar
  10. 10.
    Kimber T., “Modularity in the lattice of Σ-permutable subgroups,” Arch. Math., vol. 83, 193–203 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Schmidt R., Subgroup Lattices of Groups, Walter de Gruyter, Berlin (1994).CrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Inc. 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsJiangsu Normal UniversityXuzhouP. R. China
  2. 2.Francisk Skorina Gomel State UniversityGomelBelarus

Personalised recommendations