Algebra and Logic

, Volume 50, Issue 1, pp 17–28 | Cite as

Levi quasivarieties of exponent p s

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For an arbitrary class M of groups, L(M) denotes a class of all groups G the normal closure of any element in which belongs to M; qM is a quasivariety generated by M. Fix a prime p, p ≠ 2, and a natural number s, s ≥ 2. Let qF be a quasivariety generated by a relatively free group in a class of nilpotent groups of class at most 2 and exponent p s , with commutator subgroups of exponent p. We give a description of a Levi class generated by qF. Fix a natural number n, n ≥ 2. Let K be an arbitrary class of nilpotent groups of class at most 2 and exponent 2 n , with commutator subgroups of exponent 2. Assume also that for all groups in K, elements of order 2 m , 0 < m < n, are contained in the center of a given group. It is proved that a Levi class generated by a quasivariety qK coincides with a variety of nilpotent groups of class at most 2 and exponent 2 n , with commutator subgroups of exponent 2.

Keywords

quasivariety Levi classes nilpotent groups 

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References

  1. 1.
    L. C. Kappe, “On Levi-formations,” Arch. Math., 23, No. 6, 561–572 (1972).CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    F. W. Levi, “Groups in which the commutator operation satisfies certain algebraic conditions,” J. Indian Math. Soc., 6, 87–97 (1942).MATHMathSciNetGoogle Scholar
  3. 3.
    R. F. Morse, “Levi-properties generated by varieties,” in Cont. Math., 169, Am. Math. Soc., Providence, RI (1994), pp. 467–474.Google Scholar
  4. 4.
    A. I. Budkin, “Levi quasivarieties,” Sib. Mat. Zh., 40, No. 2, 266–270 (1999).MATHMathSciNetGoogle Scholar
  5. 5.
    A. I. Budkin, “Levi classes generated by nilpotent groups,” Algebra Logika, 39, No. 6, 635–647 (2000).MATHMathSciNetGoogle Scholar
  6. 6.
    L. C. Kappe and W. P. Kappe, “On three-Engel groups,” Bull. Austr. Math. Soc., 7, No. 3, 391–405 (1972).CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    A. I. Budkin and L. V. Taranina, “On Levi quasivarieties generated by nilpotent groups,” Sib. Mat. Zh., 41, No. 2, 270–277 (2000).MATHMathSciNetGoogle Scholar
  8. 8.
    V. V. Lodeishchikova, “On Levi quasivarieties generated by nilpotent groups,” Izv. Altai State Univ., No. 1(61), 26–29 (2009).Google Scholar
  9. 9.
    A. I. Budkin and V. A. Gorbunov, “Quasivarieties of algebraic systems,” Algebra Logika, 14, No. 2, 123–142 (1975).MATHMathSciNetGoogle Scholar
  10. 10.
    A. I. Budkin, Quasivarieties of Groups [in Russian], Altai State Univ., Barnaul (2002).Google Scholar
  11. 11.
    M. I. Kargapolov and Yu. I. Merzlyakov, Fundamentals of Group Theory [in Russian], Nauka, Moscow (1984).Google Scholar
  12. 12.
    A. I. Mal’tsev, Algebraic Systems [in Russian], Nauka, Moscow (1970).Google Scholar
  13. 13.
    H. Neumann, Varieties of Groups, Springer, Berlin (1967).MATHGoogle Scholar
  14. 14.
    A. G. Kurosh, Group Theory [in Russian], Nauka, Moscow (1967).Google Scholar
  15. 15.
    V. A. Gorbunov, Algebraic Theory of Quasivarieties, Sib. School Alg. Log. [in Russian], Nauch. Kniga, Novosibirsk (1999).Google Scholar
  16. 16.
    A. N. Fyodorov, “Quasi-identities of finite 2-nilpotent groups,” VINITI, Dep. No. 5489-B87 (1987).Google Scholar

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© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  1. 1.ul. Georgieva 4-6BarnaulRussia

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