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The Axiomatic Rank of Levi Classes

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Algebra and Logic Aims and scope

A Levi class L(ℳ) generated by a class ℳ of groups is a class of all groups in which the normal closure of each element belongs to ℳ. It is stated that there exist finite groups G such that a Levi class L(qG), where qG is a quasivariety generated by a group G, has infinite axiomatic rank. This is a solution for [The Kourovka Notebook, Quest. 15.36]. Moreover, it is proved that a Levi class L(ℳ), where ℳ is a quasivariety generated by a relatively free 2-step nilpotent group of exponent ps with a commutator subgroup of order p, p is a prime, p ≠ 2, s ≥ 2, is finitely axiomatizable.

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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).

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  3. A. I. Budkin, “Levi classes generated by nilpotent groups,” Algebra and Logic, 39, No. 6, 363-369 (2000).

    Article  MathSciNet  Google Scholar 

  4. 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 

  5. V. V. Lodeishchikova, “On Levi quasivarieties generated by nilpotent groups,” Izv. Altai State Univ., No. 1(61), 26-29 (2009).

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  8. Unsolved Problems in Group Theory, The Kourovka Notebook, No. 19, Institute of Mathematics SO RAN, Novosibirsk (2018); http://www.math.nsc.ru/∼alglog/19tkt.pdf.

  9. M. I. Kargapolov and Yu. I. Merzlyakov, Fundamentals of Group Theory [in Russian], 4th edn., Nauka, Moscow (1996).

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

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

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

    Article  MathSciNet  Google Scholar 

  13. A. Yu. Ol’shanskii, “Conditional identities in finite groups,” Sib. Math. J., 15, No. 6, 1000-1003 (1974).

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Book  Google Scholar 

Download references

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  1. Altai State University, pr. Lenina 61, Barnaul, 656049, Russia

    S. A. Shakhova

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  1. S. A. Shakhova
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Correspondence to S. A. Shakhova.

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Translated from Algebra i Logika, Vol. 57, No. 5, pp. 587-600, September-October, 2018.

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Shakhova, S.A. The Axiomatic Rank of Levi Classes. Algebra Logic 57, 381–391 (2018). https://doi.org/10.1007/s10469-018-9510-9

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  • DOI: https://doi.org/10.1007/s10469-018-9510-9

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