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Algebra Logika, 2014, Volume 53, Number 2, Pages 206–215 (Mi al631)  

This article is cited in 1 scientific paper (total in 1 paper)

Automorphisms of divisible rigid groups

D. V. Ovchinnikov

Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090, Russia

Abstract: A group $G$ is $m$-rigid if there exists a normal series of the form
$$ G=G_1>G_2>\ldots>G_m>G_{m+1}=1 $$
in which every factor $G_i/G_{i+1}$ is an Abelian group and is torsion-free as a (right) $\mathbb Z[G/G_i]$-module. A rigid group is one that is $m$-rigid for some $m$. The specified series is determined by a given rigid group uniquely; so it consists of characteristic subgroups and is called a rigid series; the solvability length of a group is exactly $m$. A rigid group $G$ is divisible if all $G_i/G_{i+1}$ are divisible modules over $\mathbb Z[G/G_i]$. The rings $\mathbb Z[G/G_i]$ satisfy the Ore condition, and $Q(G/G_i)$ denote the corresponding (right) division rings. Thus, for a divisible rigid group $G$, the factor $G_i/G_{i+1}$ can be treated as a (right) vector space over $Q(G/G_i)$.
We describe the group of all automorphisms of a divisible rigid group, and then a group of normal automorphisms. An automorphism is normal if it keeps all normal subgroups of the given group fixed.

Keywords: divisible rigid group, group of automorphisms, group of normal automorphisms.

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English version:
Algebra and Logic, 2014, 53:2, 133–139

Bibliographic databases:

UDC: 512.5
Received: 30.11.2013
Revised: 15.01.2014

Citation: D. V. Ovchinnikov, “Automorphisms of divisible rigid groups”, Algebra Logika, 53:2 (2014), 206–215; Algebra and Logic, 53:2 (2014), 133–139

Citation in format AMSBIB
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\by D.~V.~Ovchinnikov
\paper Automorphisms of divisible rigid groups
\jour Algebra Logika
\yr 2014
\vol 53
\issue 2
\pages 206--215
\mathnet{http://mi.mathnet.ru/al631}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3237390}
\transl
\jour Algebra and Logic
\yr 2014
\vol 53
\issue 2
\pages 133--139
\crossref{https://doi.org/10.1007/s10469-014-9277-6}
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\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84905028534}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. N. S. Romanovskii, “Generalized rigid groups: definitions, basic properties, and problems”, Siberian Math. J., 59:4 (2018), 705–709  mathnet  crossref  crossref  isi  elib
  • Алгебра и логика Algebra and Logic
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