On Finite Groups with Some Minimal Subgroups Weakly Supplemented

Abstract

A subgroup H of a finite group G is weakly supplemented in G if there exists a proper subgroup K of G such that \(G=HK\). In the paper, we present some sufficient and necessary conditions for a finite group to be p-nilpotent and solvable by using some weakly supplemented minimal subgroups. As applications, we extend some known results.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Arad, Z., Ward, M.B.: New criteria for the solvability of finite groups. J. Algebra 77, 234–246 (1982)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Ballester-Bolinches, A., Guo, X.: On complemented subgroups of finite groups. Arch. Math. 72, 161–166 (1999)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Hall, P.: A characteristic property of soluble groups. J. Lond. Math. Soc. 12, 188–200 (1937)

    MathSciNet  Google Scholar 

  4. 4.

    Hall, P.: Complemented groups. J. Lond. Math. Soc. 12, 201–204 (1937)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Huppert, B.: Endliche Gruppen I. Springer, Berlin (1967)

    Book  Google Scholar 

  6. 6.

    Kang, P.: Minimal subgroups and the structure of finite groups. Ric. mat. 62, 91–95 (2013)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Kong, Q., Liu, Q.: The influence of weakly-supplemented subgroups on the structure of finite groups. Czechoslov. Math. J. 64(139), 173–182 (2014)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Robinson, D.J.S.: A Course in the Theory of Groups. Springer, New York (1993)

    Google Scholar 

  9. 9.

    Rose, J.S.: On finite insolvable groups with nilpotent maximal subgroups. J. Algebra 48, 182–196 (1977)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Wei, H., Wang, Y., Liu, X.: Some necessary and sufficient conditions for \(p\)-nilpotence of finite groups. Bull. Aust. Math. Soc. 68, 371–378 (2003)

    MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Qingjun Kong.

Additional information

The paper is dedicated to Professor John Cossey for his 75th birthday.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The research of the authors is supported by the National Natural Science Foundation of China (11301378).

Communicated by Dr. Peyman Niroomand.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kong, Q., Li, J. On Finite Groups with Some Minimal Subgroups Weakly Supplemented. Bull. Malays. Math. Sci. Soc. 43, 2989–2994 (2020). https://doi.org/10.1007/s40840-019-00846-1

Download citation

Keywords

  • Weakly supplemented subgroups
  • p-nilpotent groups
  • Solvable groups

Mathematics Subject Classification

  • 20D10
  • 20D20