Czechoslovak Mathematical Journal

, Volume 58, Issue 4, pp 1083–1095 | Cite as

On S-quasinormal and c-normal subgroups of a finite group

Article
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Abstract

Let
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be a saturated formation containing the class of supersolvable groups and let G be a finite group. The following theorems are presented: (1) G ∈
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if and only if there is a normal subgroup H such that G/H ∈
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and every maximal subgroup of all Sylow subgroups of H is either c-normal or S-quasinormally embedded in G. (2) G ∈
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if and only if there is a normal subgroup H such that G/H ∈
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and every maximal subgroup of all Sylow subgroups of F*(H), the generalized Fitting subgroup of H, is either c-normal or S-quasinormally embedded in G. (3) G ∈
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if and only if there is a normal subgroup H such that G/H ∈
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and every cyclic subgroup of F*(H) of prime order or order 4 is either c-normal or S-quasinormally embedded in G.

Keywords

S-quasinormally embedded subgroup c-normal subgroup p-nilpotent group the generalized Fitting subgroup saturated formation 
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References

  1. [1]
    M. Asaad and A. A. Heliel: On S-quasinormal embedded subgroups of finite groups. J. Pure App. Algebra 165 (2001), 129–135.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    A. Ballester-Bolinches and M. C. Pedraza-Aguilera: Sufficient conditions for supersolvability of finite groups. J. Pure App. Algebra 127 (1998), 113–118.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    W. E. Deskins: On quasinormal subgroups of finite groups. Math. Z. 82 (1963), 125–132.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    O. H. Kegel: Sylow Gruppen und subnormalteiler endlicher Gruppen. Math. Z. 78 (1962), 205–221.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    X. Y. Guo and K. P. Shum: On c-normal maximal and minimal subgroups of Sylow p-subgroups of finite groups. Arch. Math. 80 (2003), 561–569.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    B. Huppert: Endliche Gruppen I. Springer-Verlag, Berlin-Heidelberg-New York, 1967.MATHGoogle Scholar
  7. [7]
    B. Huppert and N. Blackburn: Finite Groups III. Springer-Verlag, Berlin, New York, 1982.MATHGoogle Scholar
  8. [8]
    D. Li and X. Guo: The influence of c-normality of subgroups on structure of finite groups. Comm. Algebra 26 (1998), 1913–1922.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    D. Li and X. Guo: The influence of c-normality of subgroups on structure of finite groups II. J. Pure App. Algebra 150 (2000), 53–60.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    Shirong Li and Xuanli He: On normally embedded subgroups of prime power order in finite groups. Comm. Algebra 36 (2008), 2333–2340.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Yangming Li and Yanming Wang: On π-quasinormally embedded subgroups of finite group. J. Algebra 281 (2004), 109–123.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    P. Schmid: Subgroups permutable with all Sylow subgroups. J. Algebra 207 (1998), 285–293.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    S. Srinivasan: Two sufficient conditions for supersolvability of finite groups. Israel J. Math. (1980), 210–214.Google Scholar
  14. [14]
    J. Tate: Nilpotent quotient groups. Topology 3 (1964), 109–111.CrossRefMathSciNetGoogle Scholar
  15. [15]
    Yanming Wang: c-normality of groups and its properties. J. Algebra 180 (1996), 954–965.MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    H. Wei: On c-normal maximal and minimal subgroups of Sylow subgroups of finite groups. Comm. Algebra 29 (2001), 2193–2200.MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    H. Wei, Y. Wang and Y. Li: On c-normal maximal and minimal subgroups of Sylow subgroups of finite groups II. Comm. Algebra 31 (2003), 4807–4816.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Mathematical Institute, Academy of Sciences of Czech Republic 2008

Authors and Affiliations

  1. 1.Department of MathematicsGuangxi UniversityNanning, GuangxiChina
  2. 2.Department of MathematicsGuangdong College of EducationGuangzhouChina

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