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.
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Translated from Algebra i Logika, Vol. 50, No. 1, pp. 26–41, January–February, 2010.
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Lodeishchikova, V.V. Levi quasivarieties of exponent p s . Algebra Logic 50, 17–28 (2011). https://doi.org/10.1007/s10469-011-9121-1
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DOI: https://doi.org/10.1007/s10469-011-9121-1