Skip to main content
SpringerLink
Log in

On σ-semipermutable Subgroups of Finite Groups

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

Let σ = {σi|i ∈ I} be some partition of the set of all primes ℙ, G a finite group and σ(G) = {σi|σi ∩ π(G) ≠ ∅}. A set H of subgroups of G is said to be a complete Hall σ-set of G if every member ≠ 1 of H is a Hall σi-subgroup of G for some σi ∈ σ and H contains exactly one Hall σi-subgroup of G for every σi ∈ σ(G). A subgroup H of G is said to be: σ-semipermutable in G with respect to H if HH x i = H x i H for all x ∈ G and all Hi ∈ H such that (|H|, |Hi|) = 1; σ-semipermutable in G if H is σ-semipermutable in G with respect to some complete Hall σ-set of G. We study the structure of G being based on the assumption that some subgroups of G are σ-semipermutable in 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. Ballester-Bolinches, A., Ezquerro, L. M.: Classes of Finite Groups, Springer, Dordrecht, 2006

    MATH  Google Scholar 

  2. Ballester-Bolinches, A., Esteban-Romero, R., Asaad, M.: Products of Finite Groups, Walter de Gruyter, Berlin, New York, 2010

    Book  MATH  Google Scholar 

  3. Beidleman, J. C., Skiba, A. N.: On τσ-quasinormal subgroups of finite groups. J. Grou. Theory, 20(5), 955–964 (2017)

    MATH  Google Scholar 

  4. Bercovich, Ya., Isaacs, I. M.: p-supersolvability and actions on p-groups stabilizing certain subgroups. J. Algebra, 414, 82–94 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Borovikov, M. T.: Groups with permutable subgroups of mutually simple orders. Problem. Alg., 5, 80–82 (1990)

    MATH  Google Scholar 

  6. Buckley, J.: Finite groups whose minimal subgroups are normal. Math. Z., 116, 15–17 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, X., Guo,W., Skiba, A. N.: Some conditions under which a finite group belongs a Baer local formation. Comm. Algebra, 42, 4188–4205 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Doerk, K., Hawkes, T.: Finite Soluble Groups, Walter de Gruyter, Berlin, New York, 1992

    Book  MATH  Google Scholar 

  9. Gorenstein, D.: Finite Groups, Harper & Row Publishers, New York, Evanston, London, 1968

    Google Scholar 

  10. Guo, W.: Structure Theory for Canonical Classes of Finite Groups, Springer, Heidelberg, 2015

    Book  MATH  Google Scholar 

  11. Guo, W., Skiba, A. N.: On Π-quasinormal subgroups of finite groups. Monatsh. Math., DOI 10.1007/s00605-016-1007-9

  12. Guo, W., Skiba, A. N.: On σ-semipermutable subgroups of finite groups, Preprint (2016), http://arxiv.org/abs/1609.08815v1

    Google Scholar 

  13. Guo, W., Skiba, A. N.: Groups with maximal subgroups of Sylow subgroups σ-permutably embedded. J. Grou. Theory, 20, 169–183 (2017)

    MathSciNet  MATH  Google Scholar 

  14. Guo, W., Shum, K. P., Skiba, A. N.: Schur–Zassenhaus theorem forX-permutable subgroups. Algebr. Colloquium, 15, 185–192 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Guo, W., Shum, K. P., Skiba, A. N.: Finite groups with some given systems of X m-semipermutable subgroups. Math. Nachr., 283, 1603–1612 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Guo, W., Cao, C., Skiba, A. N., et al.: Finite groups with H-permutable subgroups. Commun. Math. Stat., 5(1), 83–92 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hu, B., Huang, J., Skiba, A. N.: Finite groups with given systems of σ-semipermutable subgroups. J. Algebra Appl., 17, 1850031, 13 pages (2018)

    Article  MathSciNet  MATH  Google Scholar 

  18. Huppert, B.: Endliche Gruppen I, Springer-Verlag, Berlin, Heidelberg, New York, 1967

    Book  Google Scholar 

  19. Huppert, B.: Zur Sylow struktur Auflösbarer Gruppen. Arch. Math., 12, 161–169 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  20. Isaacs, I. M.: Semipermutable π-subgroups. Arch. Math. (Basel), 102, 1–6 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kegel, O. H.: Produkte nilpotenter Gruppen. Arch. Math., 12, 90–93 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  22. Knyagina, B. N., Monakhov, V. S.: On π′-properties of finite groups having a Hall p-subgroup. Siberian Math. J., 522, 398–309 (2011)

    Google Scholar 

  23. Lennox, J. C., Stonehewer, S. E.: Subnormal Subgroups of Groups, Clarendon Press, Oxford, 1987

    MATH  Google Scholar 

  24. Li, S., Shen, Z., Kong, X.: On SS-quasinormal subgroups of finite groups. Comm. Algebra, 36, 4436–4447 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Li, S., Shen, Z., Liu, J., et al.: The influence of SS-quasinormality of some subgroups on the structure of finite group. J. Algebra, 319, 4275–4287 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Li, Y., Li, X., Wang, Y.: On s-semipermutable subgroups of finite groups. Acta Math. Sin., Engl. Ser., 26(11), 2215–2222 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Sergienko, V. I.: A criterion for sulubility of finite groups. Mat. Zam., 9, 375–383 (1971) (Russian, English translation in Math. Notes, 9, 216–220 (1971))

    MATH  Google Scholar 

  28. Skiba, A. N.: A generalization of a Hall theorem. J. Algebr. Appl., 15(4), 21–36 (2015)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  30. Skiba, A. N.: A characterization of hypercyclically embedded subgroups of finite groups. J. Pure Applie. Algebra, 215, 257–261 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. Wei, X., Guo, X.: On SS-quasinormal subgroups and the structure of finite groups. Sci. Chin. Math., 54(3), 449–456 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  32. Weinstein, M., ed.: Between Nilpotent and Solvable, Polygonal Publishing House, Passaic, NJ, 1982

Download references

Acknowledgements

The authors cordially thank the referees for their careful reading and helpful comments.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Science and Technology of China, Hefei, 230026, P. R. China

    Wen Bin Guo

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

    Alexander N. Skiba

Authors
  1. Wen Bin Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexander N. Skiba
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wen Bin Guo.

Additional information

Supported by NNSF (Grant No. 11771409) and Wu Wen-Tsun Key Laboratory of Mathematics of Chinese Academy of Sciences

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guo, W.B., Skiba, A.N. On σ-semipermutable Subgroups of Finite Groups. Acta. Math. Sin.-English Ser. 34, 1379–1390 (2018). https://doi.org/10.1007/s10114-018-6428-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-018-6428-z

Keywords

MR(2010) Subject Classification

Search

Navigation