SpringerLink
Log in

Characterizations of Finite \(\sigma \)-Nilpotent and \(\sigma \)-Quasinilpotent Groups

  • Published:
Bulletin of the Malaysian Mathematical Sciences Society Aims and scope Submit manuscript

Abstract

Let \(\sigma =\{\sigma _{i} | i\in I\}\) be a partition of the set \(\mathbb {P}\) of all primes and G a finite group. A chief factor H / K of G is said to be \(\sigma \)-central (in G) if the semidirect product \((H/K) \rtimes (G/C_{G}(H/K))\) is a \(\sigma _{i}\)-group for some \(i=i(H/K)\); otherwise, it is called \(\sigma \)-eccentric. We say that G is: \(\sigma \)-nilpotent if every chief factor of G is \(\sigma \)-central; \(\sigma \)-quasinilpotent if for every \(\sigma \)-eccentric chief factor H / K of G, every automorphism of H / K induced by an element of G is inner. In this paper, we study properties of \(\sigma \)-nilpotent and \(\sigma \)-quasinilpotent subgroups of finite groups.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Skiba, A.N.: On \(\sigma \)-subnormal and \(\sigma \)-permutable subgroups of finite groups. J. Algebra 436, 1–16 (2015)

    Article  MathSciNet  Google Scholar 

  2. Shemetkov, L.A.: Formations of Finite Groups. Nauka, main editorial board for physical and mathematical literature, Moscow (1978)

    MATH  Google Scholar 

  3. Guo, W., Skiba, A.N.: Finite groups with permutable complete Wielandt sets of subgroups. J. Group Theory 18, 191–200 (2015)

    Article  MathSciNet  Google Scholar 

  4. Ballester-Bolinches, A., Doerk, K., Pèrez-Ramos, M.D.: On the lattice of \(\mathfrak{F}\)-subnormal subgroups. J. Algebra 148, 42–52 (1992)

    Article  MathSciNet  Google Scholar 

  5. Vasil’ev, A.F., Kamornikov, A.F., Semenchuk, V.N.: On lattices of subgroups of finite groups. In: Chernikov, N.S. (ed.) Infinite Groups and Related Algebraic Structures, pp. 27–54. Institut Matemayiki AN Ukrainy, Kiev (1993). (in Russian)

    MATH  Google Scholar 

  6. Ballester-Bolinches, A., Ezquerro, L.M.: Classes of Finite Groups. Springer, Dordrecht (2006)

    MATH  Google Scholar 

  7. Beidleman, J.C., Skiba, A.N.: On \(\tau _{\sigma }\) -quasinormal subgroups of finite groups. J. Group Theory (2017). https://doi.org/10.1515/jgth-2017-0016

    Article  Google Scholar 

  8. Al-Sharo, K.A., Skiba, A.N.: On finite groups with \(\sigma \)-subnormal Schmidt subgroups. Commun. Algebra 45(10), 4158–4165 (2017)

    Article  MathSciNet  Google Scholar 

  9. Huang, J., Hu, B., Wu, X.: Finite groups all of whose subgroups are \(\sigma \)-subnormal or \(\sigma \)-abnormal. Commun. Algebra 45(10), 4542–4549 (2017)

    Article  MathSciNet  Google Scholar 

  10. Skiba, A.N.: Some characterizations of finite \(\sigma \)-soluble \(P\sigma T\)-groups. J. Algebra 495(1), 114–129 (2018). https://doi.org/10.1016/j.jalgebra.2017.11.009

    Article  MathSciNet  MATH  Google Scholar 

  11. Skiba, A.N.: On some results in the theory of finite partially soluble groups. Commun. Math. Stat. 4(3), 281–309 (2016)

    Article  MathSciNet  Google Scholar 

  12. Huppert, B., Blackburn, N.: Finite Groups III. Springer, Berlin (1982)

    Book  Google Scholar 

  13. Shemetkov, L.A.: Composition formations and radicals of finite groups. Ukrainian. Math. J. 40(3), 369–374 (1988)

    MathSciNet  MATH  Google Scholar 

  14. Doerk, K., Hawkes, T.: Finite Soluble Groups. Walter de Gruyter, Berlin (1992)

    Book  Google Scholar 

  15. Huppert, B., Blackburn, N.: Finite Groups II. Springer, Berlin (1982)

    Book  Google Scholar 

  16. Derr, J.B., Deskins, W.E., Mukherjee, N.P.: The influence of minimal \(p\)-subgroups on the structure of finite groups. Arch. Math. 45, 1–4 (1985)

    Article  MathSciNet  Google Scholar 

  17. Terence, M.: Gagen, Topics in Finite Groups. London Math Soc. Lecture Note Series, vol. 16. Cambridge University Press, Cambridge (1976)

    Google Scholar 

  18. Skiba, A.N., Shemetkov, L.A.: Multiply \(\mathfrak{L}\)-composition formations of finite groups. Ukrainian. Math. J. 52(6), 898–913 (2000)

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

Download references

Acknowledgements

The authors are deeply grateful to the helpful suggestions of the referees.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, People’s Republic of China

    Bin Hu & Jianhong Huang

  2. Department of Mathematics and Technologies of Programming, Francisk Skorina Gomel State University, 246019, Gomel, Belarus

    Alexander N. Skiba

Authors
  1. Bin Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jianhong Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alexander N. Skiba
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jianhong Huang.

Additional information

Communicated by Kar ** Shum.

Research is supported by an NNSF Grant of China (Grant No. 11401264) and a TAPP of Jiangsu Higher Education Institutions (PPZY 2015A013).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hu, B., Huang, J. & Skiba, A.N. Characterizations of Finite \(\sigma \)-Nilpotent and \(\sigma \)-Quasinilpotent Groups. Bull. Malays. Math. Sci. Soc. 42, 2091–2104 (2019). https://doi.org/10.1007/s40840-017-0593-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40840-017-0593-6

Keywords

Mathematics Subject Classification

Search

Navigation