Czechoslovak Mathematical Journal

, Volume 65, Issue 2, pp 427–433 | Cite as

On solvability of finite groups with some ss-supplemented subgroups

  • Jiakuan Lu
  • Yanyan Qiu


A subgroup H of a finite group G is said to be ss-supplemented in G if there exists a subgroup K of G such that G = HK and HK is s-permutable in K. In this paper, we first give an example to show that the conjecture in A.A. Heliel’s paper (2014) has negative solutions. Next, we prove that a finite group G is solvable if every subgroup of odd prime order of G is ss-supplemented in G, and that G is solvable if and only if every Sylow subgroup of odd order of G is ss-supplemented in G. These results improve and extend recent and classical results in the literature.


ss-supplemented subgroup solvable group supersolvable group 

MSC 2010

20D10 20D20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Z. Arad, M. B. Ward: New criteria for the solvability of finite groups. J. Algebra 77 (1982), 234–246.MATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    M. Asaad, M. Ramadan: Finite groups whose minimal subgroups are c-supplemented. Commun. Algebra 36 (2008), 1034–1040.MATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    A. Ballester-Bolinches, Y. Wang, G. Xiuyun: c-supplemented subgroups of finite groups. Glasg. Math. J. 42 (2000), 383–389.MATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    A. Ballester-Bolinches, G. Xiuyun: On complemented subgroups of finite groups. Arch. Math. 72 (1999), 161–166.MATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    K. Doerk, T. O. Hawkes: Finite Soluble Groups. de Gruyter Expositions in Mathematics 4, Walter de Gruyter, Berlin, 1992.MATHCrossRefGoogle Scholar
  6. [6]
    D. Gorenstein: Finite Groups. Harper’s Series in Modern Mathematics, Harper & Row, Publishers, New York, 1968.Google Scholar
  7. [7]
    X. Guo, J. Lu: On ss-supplemented subgroups of finite groups and their properties. Glasg. Math. J. 54 (2012), 481–491.MathSciNetCrossRefGoogle Scholar
  8. [8]
    R. M. Guralnick: Subgroups of prime power index in a simple group. J. Algebra 81 (1983), 304–311.MATHMathSciNetCrossRefGoogle Scholar
  9. [9]
    P. Hall: A characteristic property of soluble groups. J. Lond. Math. Soc. 12 (1937), 198–200.CrossRefGoogle Scholar
  10. [10]
    P. Hall: Complemented groups. J. Lond. Math. Soc. 12 (1937), 201–204.CrossRefGoogle Scholar
  11. [11]
    A. A. Heliel: A note on c-supplemented subgroups of finite groups. Commun. Algebra 42 (2014), 1650–1656.MATHMathSciNetCrossRefGoogle Scholar
  12. [12]
    B. Huppert: Endliche Gruppen. I. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 134, Springer, Berlin, 1967. (In German.)MATHGoogle Scholar
  13. [13]
    O. H. Kegel: Sylow-Gruppen und Subnormalteiler endlicher Gruppen. Math. Z. 78 (1962), 205–221. (In German.)MATHMathSciNetCrossRefGoogle Scholar
  14. [14]
    S. Li, Z. Shen, J. Liu, X. Liu: The influence of ss-quasinormality of some subgroups on the structure of finite groups. J. Algebra 319 (2008), 4275–4287.MATHMathSciNetCrossRefGoogle Scholar
  15. [15]
    Y. Li, B. Li: On minimal weakly s-supplemented subgroups of finite groups. J. Algebra Appl. 10 (2011), 811–820.MATHMathSciNetCrossRefGoogle Scholar
  16. [16]
    J. Lu, X. Guo, X. Li: The influence of minimal subgroups on the structure of finite groups. J. Algebra Appl. 12 (2013), Article No. 1250189, 8 pages.Google Scholar
  17. [17]
    P. Schmid: Subgroups permutable with all Sylow subgroups. J. Algebra 207 (1998), 285–293.MATHMathSciNetCrossRefGoogle Scholar
  18. [18]
    Y. Wang: Finite groups with some subgroups of Sylow subgroups c-supplemented. J. Algebra 224 (2000), 467–478.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2015

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsGuangxi Normal UniversityGuilinGuangxi, P.R. China

Personalised recommendations