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

This article is cited in 17 scientific papers (total in 17 papers)

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
Received: 02.08.2010

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
\by N.~S.~Romanovskii
\paper Coproducts of rigid groups
\jour Algebra Logika
\yr 2010
\vol 49
\issue 6
\pages 803--818
\jour Algebra and Logic
\yr 2010
\vol 49
\issue 6
\pages 539--550

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