, Volume 50, Issue 1, pp 17-28
Date: 02 Apr 2011

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.

Translated from Algebra i Logika, Vol. 50, No. 1, pp. 26–41, January–February, 2010.