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Algebra and Logic

, Volume 57, Issue 5, pp 381–391 | Cite as

The Axiomatic Rank of Levi Classes

  • S. A. Shakhova
Article
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A Levi class L(ℳ) generated by a class ℳ of groups is a class of all groups in which the normal closure of each element belongs to 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 [The Kourovka Notebook, Quest. 15.36]. Moreover, it is proved that a Levi class L(ℳ), where ℳ is a quasivariety generated by a relatively free 2-step nilpotent group of exponent ps with a commutator subgroup of order p, p is a prime, p ≠ 2, s ≥ 2, is finitely axiomatizable.

Keywords

quasivariety nilpotent group Levi class axiomatic rank 

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Altai State UniversityBarnaulRussia

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