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On Strongly Π-Permutable Subgroups of a Finite Group

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Abstract

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.

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References

  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).

    Article  MathSciNet  MATH  Google Scholar 

  2. Skiba A. N., “On σ-subnormal and σ-permutable subgroups of finite groups,” J. Algebra, vol. 436, 1–16 (2015).

    Article  MathSciNet  Google Scholar 

  3. Skiba A. N., “Some characterizations of finite σ-soluble PσT-groups,” J. Algebra, vol. 495, 114–129 (2018).

    Article  MathSciNet  Google Scholar 

  4. Guo W. and Skiba A. N., “On Π-quasinornial subgroups of finite groups,” Monatsh. Math., vol. 185, 443–453 (2018).

    Article  MathSciNet  Google Scholar 

  5. Kegel O. H., “Sylow-Gruppen und Subnormalteiler endlicher Gruppen,” Math. Z., Bd 78, 205–221 (1962).

    Article  MathSciNet  MATH  Google Scholar 

  6. Ballester-Bolinches A., Esteban-Romero R., and Asaad M., Products of Finite Groups, Walter de Gruyter, Berlin and New York (2010).

    Book  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  8. Skiba A. N., “On finite groups for which the lattice of S-permutable subgroups is distributive,” Arch. Math., vol. 109, 9–17 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  9. Doerk K. and Hawkes T., Finite Soluble Groups, Walter de Gruyter, Berlin and New York (1992).

    Book  MATH  Google Scholar 

  10. Kimber T., “Modularity in the lattice of Σ-permutable subgroups,” Arch. Math., vol. 83, 193–203 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  11. Schmidt R., Subgroup Lattices of Groups, Walter de Gruyter, Berlin (1994).

    Book  MATH  Google Scholar 

Download references

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Correspondence to J. Huang.

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Russian Text © The Author(s), 2019, published in Sibirskii Matematicheskii Zhurnal, 2019, Vol. 60, No. 4, pp. 922–932.

The authors were supported by the NNSF of China (Grant 11401264) and a TAPP of Jiangsu Higher Education Institutions (Grant PPZY 2015A013).

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Hu, B., Huang, J. & Skiba, A.N. On Strongly Π-Permutable Subgroups of a Finite Group. Sib Math J 60, 720–726 (2019). https://doi.org/10.1134/S0037446619040177

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  • DOI: https://doi.org/10.1134/S0037446619040177

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