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 Algebra Logika, 2014, Volume 53, Number 2, Pages 162–177 (Mi al628)  Rigid metabelian pro-$p$-groups

S. G. Afanas'evaa, N. S. Romanovskiiab

a Sobolev Institute of Mathematics, pr. Akad. Koptyuga 4, Novosibirsk, 630090, Russia
b Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090, Russia

Abstract: A metabelian pro-$p$-group $G$ is rigid if it has a normal series of the form
$$G=G_1\ge G_2\ge G_3=1$$
such that the factor group $A=G/G_2$ is torsion-free Abelian and $C=G_2$ is torsion-free as a $\mathbb Z_pA$-module. If $G$ is a non-Abelian group, then the subgroup $G_2$, as well as the given series, is uniquely defined by the properties mentioned. An Abelian pro-$p$-group is rigid if it is torsion-free, and as $G_2$ we can take either the trivial subgroup or the entire group. We prove that all rigid $2$-step solvable pro-$p$-groups are mutually universally equivalent.
Rigid metabelian pro-$p$-groups can be treated as $2$-graded groups with possible gradings $(1,1)$, $(1,0)$, and $(0,1)$. If a group is $2$-step solvable, then its grading is $(1,1)$. For an Abelian group, there are two options: namely, grading $(1,0)$, if $G_2=1$, and grading $(0,1)$ if $G_2=G$. A morphism between $2$-graded rigid pro-$p$-groups is a homomorphism $\varphi\colon G\to H$ such that $G_i\varphi\le H_i$. It is shown that in the category of $2$-graded rigid pro-$p$-groups, a coproduct operation exists, and we establish its properties.

Keywords: rigid metabelian pro-$p$-group, $2$-graded group. Full text: PDF file (196 kB) References: PDF file   HTML file

English version:
Algebra and Logic, 2014, 53:2, 102–113 Bibliographic databases:   UDC: 512.5

Citation: S. G. Afanas'eva, N. S. Romanovskii, “Rigid metabelian pro-$p$-groups”, Algebra Logika, 53:2 (2014), 162–177; Algebra and Logic, 53:2 (2014), 102–113 Citation in format AMSBIB
\Bibitem{AfaRom14} \by S.~G.~Afanas'eva, N.~S.~Romanovskii \paper Rigid metabelian pro-$p$-groups \jour Algebra Logika \yr 2014 \vol 53 \issue 2 \pages 162--177 \mathnet{http://mi.mathnet.ru/al628} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3237387} \transl \jour Algebra and Logic \yr 2014 \vol 53 \issue 2 \pages 102--113 \crossref{https://doi.org/10.1007/s10469-014-9274-9} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000339821300002} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84905040338} 

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• http://mi.mathnet.ru/eng/al/v53/i2/p162

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This publication is cited in the following articles:
1. 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    2. N. S. Romanovskii, “Algebraic sets in a finitely generated rigid $2$-step solvable pro-$p$-group”, Algebra and Logic, 54:6 (2016), 478–488     3. N. S. Romanovskii, “Partially divisible completions of rigid metabelian pro-$p$-groups”, Algebra and Logic, 55:5 (2016), 376–386    •  Number of views: This page: 224 Full text: 53 References: 31 First page: 16 Contact us: math-net2020_01 [at] mi-ras ru Terms of Use Registration Logotypes © Steklov Mathematical Institute RAS, 2020