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Laws of the lattice of all \({\mathfrak {X}}\)-local formations of finite groups

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Abstract

Let \({\mathfrak {X}}\) be a class of simple groups with a completeness property \(\pi ({\mathfrak {X}}) = \mathrm {char} \, {\mathfrak {X}}\). A formation is a class of finite groups closed under taking homomorphic images and finite subdirect products. Förster introduced the concept of \({\mathfrak {X}}\)-local formation in order to obtain a common extension of well-known Gaschütz–Lubeseder–Schmid, and Baer theorems (Publ Mat Univ Autònoma Barcelona 29(2–3):39–76, 1985). In the present paper, it is shown that any law of the lattice of all formations is true in the lattice of all \({\mathfrak {X}}\)-local formations.

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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., Shemetkov, L.A.: On partially saturated formations of finite groups. Sb. Math. 198(6), 757–775 (2007); translation from Mat. Sb. 198(6), 3–24 (2007) (in Russian)

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

    MATH  Google Scholar 

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

  6. Ballester-Bolinches, A., Pin, J.É., Soler-Escrivà, X.: Languages associated with saturated formations of groups. Forum Math. 27(3), 1471–1505 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Doerk, K., Hawkes, T.: Finite Soluble Groups. De Gruyter Expositions in Mathematics, vol. 4. W. de Gruyter, Berlin (1992)

    Book  MATH  Google Scholar 

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

  9. Guo, W., Skiba, A.N.: Two remarks on the identities of lattices of \(\omega \)-local and \(\omega \)-composition formations of finite groups. Russ. Math. 46(5), 12–20 (2002)

    MATH  Google Scholar 

  10. Higman, G.: Representations of general linear groups and varieties of groups. In: Proc. Int. Conf. Theory of Groups, pp. 167–173 (1965)

  11. Kostrikin A.: Around Burnside. Springer, Berlin (1990), Ergeb. Math. Grenzgeb., 3. Folge 20

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

  13. Shemetkov, L.A., Skiba, A.N., Vorob’ev, N.N.: On laws of lattices of partially saturated formations. Asian-Eur. J. Math. 2(1), 155–169 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

  15. Tsarev, A.: Algebraic lattices of partially saturated formations of finite groups. Afr. Mat. 31(3–4), 701–707 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  16. Tsarev, A.: The lattice properties of \({\mathfrak{X}}\)-local formations. J. Algebra Appl. 2150043 (2021). https://doi.org/10.1142/S0219498821500432

  17. Tsarev, A.: Laws of the lattices of \(\sigma \)-local formations of finite groups. Mediterr. J. Math. 17(3), 75 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  18. Tsarev, A.: Frattini subformations of foliated formations of \(T\)-groups. Rendiconti del Circolo Matematico di Palermo Series 2 68(3), 433–440 (2019)

  19. Tsarev, A., Vorobev, N.N.: Lattices of composition formations of finite groups and the laws. J. Algebra Appl. 17(5), 17 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  20. 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)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author thanks the anonymous referee for the careful and thoughtful reading of the note.

Funding

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1B02006812).

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

    Aleksandr Tsarev

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  1. Aleksandr Tsarev
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This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1B02006812).

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Tsarev, A. Laws of the lattice of all \({\mathfrak {X}}\)-local formations of finite groups. Ricerche mat 71, 673–680 (2022). https://doi.org/10.1007/s11587-021-00556-6

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