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 2n, with commutator subgroups of exponent 2. Assume also that for all groups in K, elements of order 2m, 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 2n, with commutator subgroups of exponent 2.
References
L. C. Kappe, “On Levi-formations,” Arch. Math., 23, No. 6, 561–572 (1972).
F. W. Levi, “Groups in which the commutator operation satisfies certain algebraic conditions,” J. Indian Math. Soc., 6, 87–97 (1942).
R. F. Morse, “Levi-properties generated by varieties,” in Cont. Math., 169, Am. Math. Soc., Providence, RI (1994), pp. 467–474.
A. I. Budkin, “Levi quasivarieties,” Sib. Mat. Zh., 40, No. 2, 266–270 (1999).
A. I. Budkin, “Levi classes generated by nilpotent groups,” Algebra Logika, 39, No. 6, 635–647 (2000).
L. C. Kappe and W. P. Kappe, “On three-Engel groups,” Bull. Austr. Math. Soc., 7, No. 3, 391–405 (1972).
A. I. Budkin and L. V. Taranina, “On Levi quasivarieties generated by nilpotent groups,” Sib. Mat. Zh., 41, No. 2, 270–277 (2000).
V. V. Lodeishchikova, “On Levi quasivarieties generated by nilpotent groups,” Izv. Altai State Univ., No. 1(61), 26–29 (2009).
A. I. Budkin and V. A. Gorbunov, “Quasivarieties of algebraic systems,” Algebra Logika, 14, No. 2, 123–142 (1975).
A. I. Budkin, Quasivarieties of Groups [in Russian], Altai State Univ., Barnaul (2002).
M. I. Kargapolov and Yu. I. Merzlyakov, Fundamentals of Group Theory [in Russian], Nauka, Moscow (1984).
A. I. Mal’tsev, Algebraic Systems [in Russian], Nauka, Moscow (1970).
H. Neumann, Varieties of Groups, Springer, Berlin (1967).
A. G. Kurosh, Group Theory [in Russian], Nauka, Moscow (1967).
V. A. Gorbunov, Algebraic Theory of Quasivarieties, Sib. School Alg. Log. [in Russian], Nauch. Kniga, Novosibirsk (1999).
A. N. Fyodorov, “Quasi-identities of finite 2-nilpotent groups,” VINITI, Dep. No. 5489-B87 (1987).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Algebra i Logika, Vol. 50, No. 1, pp. 26–41, January–February, 2010.
Rights and permissions
About this article
Cite this article
Lodeishchikova, V.V. Levi quasivarieties of exponent p s . Algebra Logic 50, 17–28 (2011). https://doi.org/10.1007/s10469-011-9121-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10469-011-9121-1