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 Algebra Logika: Year: Volume: Issue: Page: Find

 Algebra Logika, 2010, Volume 49, Number 6, Pages 803–818 (Mi al468)

Coproducts of rigid groups

N. S. Romanovskiiab

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
b Novosibirsk State University, Novosibirsk, Russia

Abstract: Let $\varepsilon=(\varepsilon_1,…,\varepsilon_m)$ be a tuple consisting of zeros and ones. Suppose that a group $G$ has a normal series of the form
$$G=G_1\ge G_2\ge…\ge G_m\ge G_{m+1}=1,$$
in which $G_i>G_{i+1}$ for $\varepsilon_i=1$, $G_i=G_{i+1}$ for $\varepsilon_i=0$, and all factors $G_i/G_{i+1}$ of the series are Abelian and are torsion free as right $\mathbb Z[G/G_i]$-modules. Such a series, if it exists, is defined by the group $G$ and by the tuple $\varepsilon$ uniquely. We call $G$ with the specified series a rigid $m$-graded group with grading $\varepsilon$. In a free solvable group of derived length $m$, the above-formulated condition is satisfied by a series of derived subgroups. We define the concept of a morphism of rigid $m$-graded groups.
It is proved that the category of rigid $m$-graded groups contains coproducts, and we show how to construct a coproduct $G\circ H$ of two given rigid $m$-graded groups. Also it is stated that if $G$ is a rigid $m$-graded group with grading $(1,1,…,1)$, and $F$ is a free solvable group of derived length $m$ with basis $\{x_1,…,x_n\}$, then $G\circ F$ is the coordinate group of an affine space $G^n$ in variables $x_1,…,x_n$ and this space is irreducible in the Zariski topology.

Keywords: rigid $m$-graded group, coproduct, coordinate group of affine space, Zariski topology.

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English version:
Algebra and Logic, 2010, 49:6, 539–550

Bibliographic databases:

UDC: 512.5

Citation: N. S. Romanovskii, “Coproducts of rigid groups”, Algebra Logika, 49:6 (2010), 803–818; Algebra and Logic, 49:6 (2010), 539–550

Citation in format AMSBIB
\Bibitem{Rom10} \by N.~S.~Romanovskii \paper Coproducts of rigid groups \jour Algebra Logika \yr 2010 \vol 49 \issue 6 \pages 803--818 \mathnet{http://mi.mathnet.ru/al468} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2829609} \transl \jour Algebra and Logic \yr 2010 \vol 49 \issue 6 \pages 539--550 \crossref{https://doi.org/10.1007/s10469-011-9116-y} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000288430700005} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-79952243956} 

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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. A. G. Myasnikov, N. S. Romanovskii, “Universal theories for rigid soluble groups”, Algebra and Logic, 50:6 (2012), 539–552
2. Romanovskiy N.S., “Presentations for Rigid Solvable Groups”, J. Group Theory, 15:6 (2012), 793–810
3. N. S. Romanovskii, “Irreducibility of an affine space in algebraic geometry over a group”, Algebra and Logic, 52:3 (2013), 262–265
4. S. G. Afanas'eva, N. S. Romanovskii, “Rigid metabelian pro-$p$-groups”, Algebra and Logic, 53:2 (2014), 102–113
5. D. V. Ovchinnikov, “Automorphisms of divisible rigid groups”, Algebra and Logic, 53:2 (2014), 133–139
6. 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
7. Myasnikov A.G. Romanovskii N.S., “Logical Aspects of the Theory of Divisible Rigid Groups”, Dokl. Math., 90:3 (2014), 697–698
8. N. S. Romanovskii, “Algebraic sets in a finitely generated rigid $2$-step solvable pro-$p$-group”, Algebra and Logic, 54:6 (2016), 478–488
9. N. S. Romanovskii, “Decomposition of a group over an Abelian normal subgroup”, Algebra and Logic, 55:4 (2016), 315–326
10. N. S. Romanovskii, “Partially divisible completions of rigid metabelian pro-$p$-groups”, Algebra and Logic, 55:5 (2016), 376–386
11. A. G. Myasnikov, N. S. Romanovskii, “Model-theoretic aspects of the theory of divisible rigid soluble groups”, Algebra and Logic, 56:1 (2017), 82–84
12. V. A. Roman'kov, “Solvability of equations in classes of solvable groups and Lie algebras”, Algebra and Logic, 56:3 (2017), 251–255
13. E. Yu. Daniyarova, A. G. Myasnikov, V. N. Remeslennikov, “Algebraic geometry over algebraic structures. VI. Geometric equivalence”, Algebra and Logic, 56:4 (2017), 281–294
14. N. S. Romanovskii, “Divisible rigid groups. Algebraic closedness and elementary theory”, Algebra and Logic, 56:5 (2017), 395–408
15. S. G. Afanas'eva, “Algebraic sets in a divisible $2$-rigid group”, Siberian Math. J., 59:2 (2018), 202–206
16. N. S. Romanovskii, “Divisible Rigid Groups. III. Homogeneity and Quantifier Elimination”, Algebra and Logic, 57:6 (2019), 478–489
17. Myasnikov A.G., Romanovskii N.S., “Characterization of Finitely Generated Groups By Types”, Int. J. Algebr. Comput., 28:8, SI (2018), 1613–1632
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