Skip to main content
SpringerLink
Log in

Supersolubility of a Finite Group with Normally Embedded Maximal Subgroups in Sylow Subgroups

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

Let P be a subgroup of a Sylow subgroup of a finite group G. If P is a Sylow subgroup of some normal subgroup of G then P is called normally embedded in G. We establish tests for a finite group G to be p-supersoluble provided that every maximal subgroup of a Sylow p-subgroup of X is normally embedded in G. We study the cases when X is a normal subgroup of G, X = Op',p(H), and X = F*(H) where H is a normal subgroup of G.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Huppert B., Endliche Gruppen I, Springer-Verlag, Berlin, Heidelberg, and New York (1967).

    Google Scholar 

  2. Monakhov V. S., Introduction to the Theory of Finite Groups and Their Classes [Russian], Vysheishaya Shkola, Minsk (2006).

    Google Scholar 

  3. Srinivasan S., “Two sufficient conditions for supersolvability of finite groups,” Israel J. Math., vol. 35, No. 3, 210–214 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  4. Wall G. L., “Groups with maximal subgroups of Sylow subgroups normal,” Israel J. Math., vol. 43, No. 2, 166–168 (1982).

    Article  MathSciNet  Google Scholar 

  5. Asaad M., Ramadan M., and Shaalan A., “Influence of π-quasinormality on maximal subgroups of Sylow subgroups of fitting subgroup of a finite group,” Arch. Math., vol. 56, 521–527 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  6. Ramadan M., “Influence of normality on maximal subgroups of Sylow subgroups of a finite group,” Acta Math. Hungar., vol. 59, No. 1–2, 107–110 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  7. Ballester-Bolinches A., “Permutably embedded subgroups of finite soluble groups,” Arch. Math., vol. 65, 1–7 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  8. Ballester-Bolinches A. and Pedraza-Aguilera M. C., “Sufficient conditions for supersolvability of finite groups,” J. Pure Appl. Algebra, vol. 127, 113–118 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  9. Asaad M. and Heliel A. A., “On S-quasinormal embedded subgroups of finite groups,” J. Pure Appl. Algebra, vol. 165, 129–135 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  10. Monakhov V. S. and Trofimuk A. A., “Finite groups with subnormal non-cyclic subgroups,” J. Group Theory, vol. 17, No. 5, 889–895 (2014).

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  12. Li S., He J., Nong G., and Zhou L., “On Hall normally embedded subgroups of finite groups,” Comm. Algebra, vol. 37, 3360–3367 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  13. Li S. and Liu J., “On Hall subnormally embedded and generalized nilpotent groups,” J. Algebra, vol. 388, 1–9 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  14. Ballester-Bolinches A. and Qiao S., “On a problem posed by S. Li and J. Liu,” Arch. Math., vol. 102, 109–111 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  15. Monakhov V. S. and Kniahina V. N., “On Hall embedded subgroups of finite groups,” J. Group Theory, vol. 18, 565–568 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  16. Monakhov V. S. and Chirik I. K., “On the p-supersolvability of a finite factorizable group with normal factors,” Trudy Inst. Mat. i Mekh. UrO RAN, vol. 21, No. 3, 256–267 (2015).

    MathSciNet  Google Scholar 

  17. GAP—Groups, Algorithms, and Programming, Version 4.8.9 (2017); http://www.gap-system.org.

  18. Monakhov V. S., “On the nilpotent π-length of maximal subgroups of finite π-soluble groups,” Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., no. 6, 3–8 (2009).

    Google Scholar 

  19. Wang Y., “C-normality of groups and its properties,” J. Algebra, vol. 180, 954–965 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  20. Wei H., “On c-normal maximal and minimal subgroups of Sylow subgroups of finite groups,” Comm. Algebra, vol. 29, No. 5, 2193–2200 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  21. Wei H., Wang Y., and Li Y., “On c-normal maximal and minimal subgroups of Sylow subgroups of finite groups. II,” Comm. Algebra, vol. 31, No. 10, 4807–4816 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  22. Huppert B. and Blackburn N., Finite Groups. III, Springer-Verlag, Berlin and New York (1982).

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Francisk Skorina Gomel State University, Gomel, Belarus

    V. S. Monakhov

  2. Pushkin Brest State University, Brest, Belarus

    A. A. Trofimuk

Authors
  1. V. S. Monakhov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. A. Trofimuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. S. Monakhov.

Additional information

Original Russian Text © 2018 Monakhov V.S. and Trofimuk A.A.

Gomel; Brest. Translated from Sibirskii Matematicheskii Zhurnal, vol. 59, no. 5, pp. 1159–1170, September–October, 2018; DOI: 10.17377/smzh.2018.59.516.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Monakhov, V.S., Trofimuk, A.A. Supersolubility of a Finite Group with Normally Embedded Maximal Subgroups in Sylow Subgroups. Sib Math J 59, 922–930 (2018). https://doi.org/10.1134/S0037446618050166

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0037446618050166

Keywords

Search

Navigation