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Algebra Logika, 2016, Volume 55, Number 5, Pages 571–586 (Mi al762)  

Partially divisible completions of rigid metabelian pro-$p$-groups

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: Previously, the author defined the concept of a rigid (abstract) group. By analogy, a metabelian pro-$p$-group $G$ is said to be rigid if it contains 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 $G_2$ being a $\mathbb Z_pA$-module is torsion-free. An abstract rigid group can be completed and made divisible. Here we do something similar for finitely generated rigid metabelian pro-$p$-groups. In so doing, we need to exit the class of pro-$p$-groups, since even the completion of a torsion-free nontrivial Abelian pro-$p$-group is not a pro-$p$-group. In order to not complicate the situation, we do not complete a first factor, i.e., the group $A$. Indeed, $A$ is simply structured: it is isomorphic to a direct sum of copies of $\mathbb Z_p$. A second factor, i.e., the group $G_2$, is completed to a vector space over a field of fractions of a ring $\mathbb Z_pA$, in which case the field and the space are endowed with suitable topologies. The main result is giving a description of coordinate groups of irreducible algebraic sets over such a partially divisible topological group.

Keywords: abstract rigid group, divisible group, coordinate group, irreducible algebraic set.

Funding Agency Grant Number
Russian Foundation for Basic Research 15-01-01485
Supported by RFBR, project No. 15-01-01485.


DOI: https://doi.org/10.17377/alglog.2016.55.504

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English version:
Algebra and Logic, 2016, 55:5, 376–386

Bibliographic databases:

UDC: 512.5
Received: 05.03.2016

Citation: N. S. Romanovskii, “Partially divisible completions of rigid metabelian pro-$p$-groups”, Algebra Logika, 55:5 (2016), 571–586; Algebra and Logic, 55:5 (2016), 376–386

Citation in format AMSBIB
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\by N.~S.~Romanovskii
\paper Partially divisible completions of rigid metabelian pro-$p$-groups
\jour Algebra Logika
\yr 2016
\vol 55
\issue 5
\pages 571--586
\mathnet{http://mi.mathnet.ru/al762}
\crossref{https://doi.org/10.17377/alglog.2016.55.504}
\transl
\jour Algebra and Logic
\yr 2016
\vol 55
\issue 5
\pages 376--386
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