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 Algebra Logika, 2016, Volume 55, Number 4, Pages 478–492 (Mi al754)

Decomposition of a group over an Abelian normal subgroup

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: Let a group $G$ have an Abelian normal subgroup $A$; put $\overline G=G/A$ and $\overline g=gA$ for $g\in G$. We can think of $A$ as a right $\mathbb Z\overline G$-module and define the action of an element $u=\alpha_1\overline g_1+…+\alpha_n\overline g_n\in\mathbb Z\overline G$ on $a\in A$ by a formula $a^u=(a^{g_1})^{\alpha_1}\cdot\ldots\cdot(a^{g_n})^{\alpha_n}$; here $a^{g_i}=g^{-1}_iag_i$. Denote by $\Theta_{\mathbb Z\overline G}(A)$ the annihilator of $A$ in the ring $\mathbb Z\overline G$, which is a two-sided ideal. Let $R=\mathbb Z\overline G/\Theta_{\mathbb Z\overline G}(A)$. A subgroup $A$ can also be treated as an $R$-module. We give a criterion for the existence of an $R$-decomposition of $G$ over $A$, i.e., the possibility of embedding $G$ in a semidirect product $\overline G\cdot D$, where $D$ is an $R$-module. It is also proved that an $R$-decomposition always exists in one important case.

Keywords: Abelian normal subgroup, $R$-decomposition.

 Funding Agency Grant Number Russian Science Foundation 14-21-00065 Supported by Russian Science Foundation, project No. 14-21-00065.

DOI: https://doi.org/10.17377/alglog.2016.55.407

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English version:
Algebra and Logic, 2016, 55:4, 315–326

Bibliographic databases:

UDC: 512.5

Citation: N. S. Romanovskii, “Decomposition of a group over an Abelian normal subgroup”, Algebra Logika, 55:4 (2016), 478–492; Algebra and Logic, 55:4 (2016), 315–326

Citation in format AMSBIB
\Bibitem{Rom16} \by N.~S.~Romanovskii \paper Decomposition of a~group over an Abelian normal subgroup \jour Algebra Logika \yr 2016 \vol 55 \issue 4 \pages 478--492 \mathnet{http://mi.mathnet.ru/al754} \crossref{https://doi.org/10.17377/alglog.2016.55.407} \transl \jour Algebra and Logic \yr 2016 \vol 55 \issue 4 \pages 315--326 \crossref{https://doi.org/10.1007/s10469-016-9401-x} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000388103400007} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84994758731} 

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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, “Divisible rigid groups. Algebraic closedness and elementary theory”, Algebra and Logic, 56:5 (2017), 395–408
2. E. I. Timoshenko, “On splittings, subgroups, and theories of partially commutative metabelian groups”, Siberian Math. J., 59:3 (2018), 536–541
3. N. S. Romanovskii, “Generalized rigid metabelian groups”, Siberian Math. J., 60:1 (2019), 148–152
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