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Levi Classes of Quasivarieties of Nilpotent Groups of Exponent ps

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Algebra and Logic Aims and scope

The Levi class L(M) generated by the class M of groups is the class of all groups in which the normal closure of every element belongs to M. It is proved that there exists a set of quasivarieties M of cardinality continuum such that \( L\left(\mathrm{M}\right)=L\left(q{H}_{p^s}\right) \), where \( q{H}_{p^s} \) is the quasivariety generated by the group \( {H}_{p^s} \), a free group of rank 2 in the variety \( {R}^{p^s} \) of ≤ 2-step nilpotent groups of exponent ps with commutator subgroup of exponent p, p is a prime number, p ≠ 2, s is a natural number, s ≥ 2, and s > 2 for p = 3.

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References

  1. F. W. Levi, “Groups in which the commutator operation satisfies certain algebraical conditions,” J. Indian Math. Soc., New Ser., 6, 87-97 (1942).

  2. R. F. Morse, “Levi-properties generated by varieties,” in Cont. Math., 169, Am. Math. Soc., Providence, RI (1994), pp. 467-474.

  3. L. C. Kappe and W. P. Kappe, “On three-Engel groups,” Bull. Austr. Math. Soc., 7, No. 3, 391-405 (1972).

    Article  MathSciNet  Google Scholar 

  4. K. W. Weston, “ZA-groups which satisfy the mth Engel condition,” Ill. J. Math., 8, No. 3, 458-472 (1964).

  5. H. Heineken, “Engelsche Elemente der L¨ange drei, Ill,” J. Math., 5, 681-707 (1961).

    MATH  Google Scholar 

  6. A. I. Budkin, “Levi quasivarieties,” Sib. Math. J., 40, No. 2, 225-228 (1999).

    Article  MathSciNet  Google Scholar 

  7. A. I. Budkin, “The operator Ln on quasivarieties of universal algebras,” Sib. Math. J., 60, No. 4, 565-571 (2019).

    Article  MathSciNet  Google Scholar 

  8. A. I. Budkin and L. V. Taranina, “On Levi quasivarieties generated by nilpotent groups,” Sib. Math. J., 41, No. 2, 218-223 (2000).

    Article  MathSciNet  Google Scholar 

  9. V. V. Lodeishchikova, “The Levi classes generated by nilpotent groups,” Sib. Math. J., 51, No. 6, 1075-1080 (2010).

    Article  MathSciNet  Google Scholar 

  10. V. V. Lodeishchikova, “Levi quasivarieties of exponent ps,” Algebra and Logic, 50, No. 1, 17-28 (2011).

    MathSciNet  MATH  Google Scholar 

  11. S. A. Shakhova, “The axiomatic rank of Levi classes,” Algebra and Logic, 57, No. 5, 381-391 (2018).

    Article  MathSciNet  Google Scholar 

  12. S. A. Shakhova, “Levi classes of quasivarieties of groups with commutator subgroup of order p,” Algebra and Logic, 60, No. 5, 336-347 (2021).

    Article  MathSciNet  Google Scholar 

  13. M. I. Kargapolov and Yu. I. Merzlyakov, Fundamentals of Group Theory [in Russian], Nauka, Moscow (1977).

    MATH  Google Scholar 

  14. H. Neumann, Varieties of Groups, Springer, Berlin (1967).

    Book  Google Scholar 

  15. A. I. Mal’tsev, Algebraic Systems [in Russian], Nauka, Moscow (1970).

  16. A. I. Budkin, Quasivarieties of Groups [in Russian], Altai State Univ., Barnaul (2002).

    MATH  Google Scholar 

  17. V. A. Gorbunov, Algebraic Theory of Quasivarieties, Sib. School Alg. Log. [in Russian], Nauch. Kniga, Novosibirsk (1999).

  18. A. I. Budkin and V. A. Gorbunov, “Quasivarieties of algebraic systems,” Algebra and Logic, 14, No. 2, 73-84 (1975).

    Article  MathSciNet  Google Scholar 

  19. S. A. Shakhova, “On the lattice of quasivarieties of nilpotent groups of class 2,” Sib. Adv. Math., 7, No. 3, 98-125 (1997).

    MathSciNet  MATH  Google Scholar 

  20. A. N. Fyodorov, “Subquasivarieties of nilpotent minimal non-Abelian group varieties,” Sib. Math. J., 21, No. 6, 840-850 (1980).

    Article  Google Scholar 

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Correspondence to V. V. Lodeishchikova.

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Translated from Algebra i Logika, Vol. 61, No. 1, pp. 77-92, January-February, 2022. Russian DOI: https://doi.org/10.33048/alglog.2022.61.104.

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Lodeishchikova, V.V., Shakhova, S.A. Levi Classes of Quasivarieties of Nilpotent Groups of Exponent ps. Algebra Logic 61, 54–66 (2022). https://doi.org/10.1007/s10469-022-09674-y

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  • DOI: https://doi.org/10.1007/s10469-022-09674-y

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