Skip to main content
SpringerLink
Log in

Algebraic lattices of partially saturated formations of finite groups

  • Published:
Afrika Matematika Aims and scope Submit manuscript

Abstract

Let \({\mathfrak {X}}\) be a class of simple groups with a completeness property \(\pi ({\mathfrak {X}}) = \mathrm {char} \, \mathfrak X\). Förster introduced the concept of \({\mathfrak {X}}\)-local formation in order to obtain a common extension of well-known theorems of Gaschütz–Lubeseder–Schmid and Baer (Publ Mat Univ Autònoma Barcelona, 29(2–3), 1985). In the present paper, it is proved that the lattice of all \({\mathfrak {X}}\)-local formations of finite groups is algebraic.

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., Calvo, C., Esteban-Romero, R.: \({\mathfrak{X}}\)-saturated formations of finite groups. Commun. Algebra 33(4), 1053–1064 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ballester-Bolinches, A., Calvo, C., Esteban-Romero, R.: Products of formations of finite groups. J. Algebra 299(2), 602–615 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ballester-Bolinches, A., Calvo, C., Esteban-Romero, R.: A question from the Kourovka notebook on formation products. Bull. Aust. Math. Soc. 68(3), 461–470 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ballester-Bolinches, A., Calvo, C., Shemetkov, L.A.: On partially saturated formations of finite groups. Sbornik: Mathematics 198(6), 757–775 (2007). translation from Matematicheskii Sbornik, 198 (6) (2007), pp. 324 (in Russian)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  6. Calvo, C.: On partially saturated formations of finite groups. Tesis doctoral, Universitat de València. Departament d’Àlgebra (2007)

  7. Chi, Z., Safonov, V.G., Skiba, A.N.: On \(n\)-multiply \(\sigma \)-local formations of finite groups. Commun. Algebra 47(3), 957–968 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chi, Z., Safonov, V.G., Skiba, A.N.: On one application of the theory of \(n\)-multiply \(\sigma \)-local formations of finite groups. Probl. Fiz. Mat. Tekh. 2(35), 85–88 (2018)

    MathSciNet  MATH  Google Scholar 

  9. Doerk, K., Hawkes, T.: Finite Soluble Groups, De Gruyter Expositions in Mathematics, vol. 4. Walter de Gruyter, Berlin, New York (1992)

    Book  MATH  Google Scholar 

  10. Förster, P.: Projektive Klassen endlicher Gruppen. Publ. Sec. Mat. Univ. Autònoma Barcelona 29(2–3), 39–76 (1985)

    MathSciNet  MATH  Google Scholar 

  11. Gaschütz, W.: Zur Theorie der endlichen auflösbaren Gruppen. Math. Z. 80(4), 300–305 (1963)

    MathSciNet  MATH  Google Scholar 

  12. Grätzer, G.: General Lattice Theory. Academic Press, Cambridge (1978)

    Book  MATH  Google Scholar 

  13. Safonov, V.G.: The property of being algebraic for the lattice of all \(\tau \)-closed totally saturated formations. Algebra. Logika 45(5), 353–356 (2006). translation from Algebra Logika 45 (5) (2006), pp. 620–626 (in Russian)

    Article  MathSciNet  Google Scholar 

  14. Scherbina, V.V., Safonov, V.G.: On some properties of the lattice of partially totally saturated formations of finite groups, Belgorod. Stat. Univ. Sci. Bull. Math. Phys. 51(2), 227–244 (2019). https://doi.org/10.18413/2075-4639-2019-51-2-227-244. (in Russian)

    Article  Google Scholar 

  15. Shabalina I.P.: Algebraic lattices of \(\tau \)-closed \(n\)-multiply \(\omega \)-local formations. Izv. Gomel. Gos. Univ. Im. F. Skoriny. Vopr. Algebry 18 5 (14), 59–67 (2002). (in Russian)

  16. Shemetkov, L.A.: Local definitions of formations of finite groups. J. Math. Sci. N Y 185(2), 324–334 (2012). translation from Fundam. Prikl. Mat., 16 (8) (2010), pp. 229–244 (in Russian)

    Article  MathSciNet  MATH  Google Scholar 

  17. Shemetkov, L.A.: Formations of Finite Groups. Sovremennaya Algebra, Nauka, Moskva (1978). (in Russian)

    MATH  Google Scholar 

  18. Shemetkov, L.A., Skiba, A.N.: Formations of Algebraic Systems. Sovremennaya Algebra, Nauka, Moskva (1989). (in Russian)

    MATH  Google Scholar 

  19. Skiba, A.N., Vorob’ev, N.N.: On the lattices of Saturated and solubly saturated formations of finite groups. Southeast Asian Bull. Math. 37(5), 771–780 (2013)

    MathSciNet  MATH  Google Scholar 

  20. Skiba, A.N.: Algebra of Formations. Belaruskaya Navuka, Minsk (1997). (in Russian)

    MATH  Google Scholar 

  21. Skiba, A.N., Shemetkov, L.A.: Multiply \({\mathfrak{L}}\)-composition formations of finite groups. Ukr. Math. J. 52(6), 898–913 (2000). translated from Ukrainskyi Matematychnyi Zhurnal, 52 (6) (2000), pp. 783–797 (in Russian)

    Article  Google Scholar 

  22. Tsarev, A.: On the lattice of all totally composition formations of finite groups. Ricerche mat. 68(2), 693–698 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  23. Tsarev, A.: On the lattice properties of \(\mathfrak{X}\)-local formations. J. Algebra Appl. (2020). https://doi.org/10.1142/S0219498821500432

  24. Tsarev, A., Vorob’ev, N.N.: Lattices of composition formations of finite groups and the laws. J. Algebra Appl. 17(5), 1850084 (2018). (17 pages)

    Article  MathSciNet  MATH  Google Scholar 

  25. Vorob’ev, N.N., Skiba, A.N.: On the distributivity of the lattice of solvable totally local Fitting classes. Math. Notes 67(5), 563–571 (2000). translation from Mat. Zametki, 67 (5) (2000), pp. 662–673 (in Russian)

    Article  MathSciNet  MATH  Google Scholar 

  26. Vorob’ev, N.N., Skiba, A.N., Tsarev, A.A.: Laws of the lattices of partially composition formations. Sib. Math. J. 52(5), 802–812 (2011). translated from Sib. Mat. Zhurnal, 52 (5) (2011), pp. 1011–1024 (in Russian)

    Article  MathSciNet  MATH  Google Scholar 

  27. Vorob’ev, N.N., Tsarev, A.A.: On the modularity of a lattice of \(\tau \)-closed \(n\)-multiply \(\omega \)-composition formations. Ukr. Math. J. 62(4), 518–529 (2010). translation from Ukrainskyi Matematychnyi Zhurnal, 62 (4) (2010), pp. 453–463 (in Russian)

    Article  Google Scholar 

Download references

Acknowledgements

This work was partially supported by the Belarusian Republican Foundation for Fundamental Research (BRFFI–RFFI M-20017, Grant F17RM-063). The author thanks the anonymous referees for many insightful comments that have greatly contributed to this paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, Jeju National University, Jeju, 690-756, South Korea

    Aleksandr Tsarev

Authors
  1. Aleksandr Tsarev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Aleksandr Tsarev.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tsarev, A. Algebraic lattices of partially saturated formations of finite groups. Afr. Mat. 31, 701–707 (2020). https://doi.org/10.1007/s13370-019-00753-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13370-019-00753-5

Keywords

Mathematics Subject Classification

Search

Navigation