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 Algebra Logika, 2010, Volume 49, Number 5, Pages 654–669 (Mi al459)

Equations and algebraic geometry over profinite groups

S. G. Melesheva

Novosibirsk State Univ., Novosibirsk, Russia

Abstract: The notion of an equation over a profinite group is defined, as well as the concepts of an algebraic set and of a coordinate group. We show how to represent the coordinate group as a projective limit of coordinate groups of finite groups. It is proved that if the set $\pi(G)$ of prime divisors of the profinite period of a group $G$ is infinite, then such a group is not Noetherian, even with respect to one-variable equations. For the case of Abelian groups, the finiteness of a set $\pi(G)$ gives rise to equational Noetherianness. The concept of a standard linear pro-$p$-group is introduced, and we prove that such is always equationally Noetherian. As a consequence, it is stated that free nilpotent pro-$p$-groups and free metabelian pro-$p$-groups are equationally Noetherian. In addition, two examples of equationally non-Noetherian pro-$p$-groups are constructed. The concepts of a universal formula and of a universal theory over a profinite group are defined. For equationally Noetherian profinite groups, coordinate groups of irreducible algebraic sets are described using the language of universal theories and the notion of discriminability.

Keywords: profinite group, equation, equationally Noetherian group, universal theory over profinite group.

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English version:
Algebra and Logic, 2010, 49:5, 444–455

Bibliographic databases:

UDC: 512.542
Revised: 30.05.2010

Citation: S. G. Melesheva, “Equations and algebraic geometry over profinite groups”, Algebra Logika, 49:5 (2010), 654–669; Algebra and Logic, 49:5 (2010), 444–455

Citation in format AMSBIB
\Bibitem{Mel10} \by S.~G.~Melesheva \paper Equations and algebraic geometry over profinite groups \jour Algebra Logika \yr 2010 \vol 49 \issue 5 \pages 654--669 \mathnet{http://mi.mathnet.ru/al459} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2796491} \zmath{https://zbmath.org/?q=an:1255.20043} \elib{http://elibrary.ru/item.asp?id=15483588} \transl \jour Algebra and Logic \yr 2010 \vol 49 \issue 5 \pages 444--455 \crossref{https://doi.org/10.1007/s10469-010-9108-3} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000288429600005} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-78650330932} 

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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. G. Afanas'eva, N. S. Romanovskii, “Rigid metabelian pro-$p$-groups”, Algebra and Logic, 53:2 (2014), 102–113
2. S. G. Afanas'eva, “The coordinate group of an affine space over a rigid metabelian pro-$p$-group”, Algebra and Logic, 53:3 (2014), 187–190
3. N. S. Romanovskii, “Algebraic sets in a finitely generated rigid $2$-step solvable pro-$p$-group”, Algebra and Logic, 54:6 (2016), 478–488
4. N. S. Romanovskii, “Partially divisible completions of rigid metabelian pro-$p$-groups”, Algebra and Logic, 55:5 (2016), 376–386
5. Casals-Ruiz M., Kazachkov I., Remeslennikov V., “Pro-Hall R-Groups and Groups Discriminated By the Free Pro-P Group”, J. Group Theory, 19:3 (2016), 391–403
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