Skip to main content
Log in

On the Distributivity and Modularity Properties of the Lattice of Fitting Classes

  • Research Articles
  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

It is proved that the lattice of all Fitting classes of finite groups is not distributive, and conditions under which Fitting classes satisfy the distributive and modular laws are determined.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. K. Doerk and T. Hawkes, Finite Soluble Groups, in De Gruyter Exp. Math. (Walter de Gruyter, Berlin, 1992), Vol. 4.

    Book  Google Scholar 

  2. D. Blessenohl and W. Gaschütz, “Über normale Schunk- und Fittingklassen,” Math. Z. 118, 1–8 (1970).

    Article  MathSciNet  Google Scholar 

  3. H. Lausch, “On normal Fitting classes,” Math. Z. 130, 67–72 (1973).

    Article  MathSciNet  Google Scholar 

  4. N. T. Vorob’ev and A. V. Martsinkevich, “Finite \(\pi\)-groups with normal injectors,” Siberian Math. J. 56 (4), 624–630 (2015).

    Article  MathSciNet  Google Scholar 

  5. N. N. Vorob’ev and A. N. Skiba, “On the distributivity of the lattice of solvable totally local Fitting classes,” Math. Notes 67 (5), 563–571 (2000).

    Article  MathSciNet  Google Scholar 

  6. S. Reifferscheid, On the Theory of Fitting Classes of Finite Soluble Groups, Dissertation (Tübingen, 2001).

  7. A. N. Skiba, “On one generalization of the local formations,” PFMT, No. 1 (34), 79–82 (2018).

    MathSciNet  MATH  Google Scholar 

  8. A. N. Skiba, “A generalization of a Hall theorem,” J. Algebra Appl. 15 (5) (2015) Paper no. 1650085.

    MathSciNet  Google Scholar 

  9. A. N. Skiba, “Some characterizations of finite \(\sigma\)-soluble \(P\sigma T\)-groups,” J. Algebra 495, 114–129 (2018).

    Article  MathSciNet  Google Scholar 

  10. A. N. Skiba, “On sublattices of the subgroup lattice defined by formation Fitting sets,” J. Algebra 550, 69–85 (2020).

    Article  MathSciNet  Google Scholar 

  11. W. Guo, L. Zhang, and N. T. Vorob’ev, “On \(\sigma\)-local Fitting classes,” J. Algebra 542 (15), 116–129 (2020).

    Article  MathSciNet  Google Scholar 

  12. A. N. Skiba, Algebra of Formations (Belaruskaya navuka, Minsk, 1997) [in Russian].

    MATH  Google Scholar 

  13. in The Kourovka Notebook. Unsolved Questions in Group Theory (Ross. Akad. Nauk, Sibirsk. Otd., Inst Mat. im. S. L. Soboleva, Novosibirsk, 1999) [in Russian].

  14. F. P. Lockett, “The Fitting class \(\mathfrak{F}^{\ast}\),” Math. Z. 137, 131–136 (1974).

    Article  MathSciNet  Google Scholar 

  15. L. A. Shemetkov, Formations of Finite Groups (Nauka, Moscow, 1978) [in Russian].

    MATH  Google Scholar 

  16. E. Cusack, “The join of two Fitting classes,” Math. Z. 167, 37–47 (1979).

    Article  MathSciNet  Google Scholar 

  17. N. Yang, Sh. Zhao, A. V. Martsinkevich, and N. T. Vorob’ev, “The sublattice of the lattice of \(\pi\)-normal Fitting classes,” J. Algebra Appl. 19 (6) (2020) Paper no. 2050105.

    Article  MathSciNet  Google Scholar 

  18. A. V. Martsinkevich and N. T. Vorob’ev, “Products and unions of locally normal Fitting classes,” in Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk (2018), Vol. 24, pp. 152–157.

    Google Scholar 

  19. A. R. Camina, “A Note on Fitting Classes,” Math. Z. 136, 351–352 (1974).

    Article  MathSciNet  Google Scholar 

  20. N. T. Vorob’ev and E. D. Lantsetova, “On lattice properties of Fitting classes,” Vesnik Vitsebsk. Dzyarzh. Univ. 98 (1), 5–10 (2017).

    Google Scholar 

  21. N. T. Vorob’ev, “On Hawkes’s conjecture for radical classes,” Siberian Math. J. 37 (6), 1137–1142 (1996).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to N. T. Vorob’ev.

Additional information

Translated from Matematicheskie Zametki, 2021, Vol. 110, pp. 658–671 https://doi.org/10.4213/mzm13135.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vorob’ev, N.T., Lantsetova, E.D. On the Distributivity and Modularity Properties of the Lattice of Fitting Classes. Math Notes 110, 655–665 (2021). https://doi.org/10.1134/S000143462111002X

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S000143462111002X

Keywords

Navigation