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 Algebra Logika, 2014, Volume 53, Number 2, Pages 206–215 (Mi al631)  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. Full text: PDF file (153 kB) References: PDF file   HTML file

English version:
Algebra and Logic, 2014, 53:2, 133–139 Bibliographic databases:   UDC: 512.5
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
\Bibitem{Ovc14} \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} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000339821300005} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84905028534} 

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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     •  Number of views: This page: 173 Full text: 43 References: 28 First page: 16 Contact us: math-net2021_11 [at] mi-ras ru Terms of Use Registration to the website Logotypes © Steklov Mathematical Institute RAS, 2021