The axiomatic rank of Levi classes. (English. Russian original) Zbl 1406.20035
Algebra Logic 57, No. 5, 381-391 (2018); translation from Algebra Logika 57, No. 5, 587-600 (2018).
Summary: A Levi class \(L(\mathcal{M})\) generated by a class \(\mathcal{M}\) of groups is a class of all groups in which the normal closure of each element belongs to \(\mathcal{M}\). 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 [V. D. Mazurov (ed.) and E. I. Khukhro (ed.), The Kourovka notebook. Unsolved problems in group theory. 19th edition. Novosibirsk: Institute of Mathematics, Russian Academy of Sciences, Siberian Div. (2018), Quest. 15.36]. Moreover, it is proved that a Levi class \(L(\mathcal{M})\), where \(\mathcal{M}\) is a quasivariety generated by a relatively free 2-step nilpotent group of exponent \(p^s\) with a commutator subgroup of order \(p\), \(p\) is a prime, \(p\neq 2\), \(s\geq2\), is finitely axiomatizable.
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