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 Algebra Logika, 2011, Volume 50, Number 1, Pages 26–41 (Mi al473)

Levi quasivarieties of exponent $p^s$

V. V. Lodeishchikova

Barnaul, Russia

Abstract: 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 $p$, $p\ne2$, and a natural number $s$, $s\ge2$. 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 $n$, $n\ge2$. 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 $2$.

Keywords: quasivariety, Levi classes, nilpotent groups.

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English version:
Algebra and Logic, 2011, 50:1, 17–28

Bibliographic databases:

UDC: 512.54.01

Citation: V. V. Lodeishchikova, “Levi quasivarieties of exponent $p^s$”, Algebra Logika, 50:1 (2011), 26–41; Algebra and Logic, 50:1 (2011), 17–28

Citation in format AMSBIB
\Bibitem{Lod11} \by V.~V.~Lodeishchikova \paper Levi quasivarieties of exponent~$p^s$ \jour Algebra Logika \yr 2011 \vol 50 \issue 1 \pages 26--41 \mathnet{http://mi.mathnet.ru/al473} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2848732} \zmath{https://zbmath.org/?q=an:1266.20040} \transl \jour Algebra and Logic \yr 2011 \vol 50 \issue 1 \pages 17--28 \crossref{https://doi.org/10.1007/s10469-011-9121-1} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000289376600002} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-79954421448} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. S. A. Shakhova, “The axiomatic rank of Levi classes”, Algebra and Logic, 57:5 (2018), 381–391
2. V. V. Lodeishchikova, “A Levi class generated by a quasivariety of nilpotent groups”, Algebra and Logic, 58:4 (2019), 327–336
3. A. I. Budkin, “The operator $L_n$ on quasivarieties of universal algebras”, Siberian Math. J., 60:4 (2019), 565–571
4. Shakhova S.A., “the Axiomatic Rank of the Levi Class Generated By the Almost Abelian Quasivariety of Nilpotent Groups”, Lobachevskii J. Math., 41:9, SI (2020), 1680–1683
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