
Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2016, Number 3(41), Pages 42–50
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This article is cited in 4 scientific papers (total in 4 papers)
MATHEMATICS
Fully inert subgroups of completely decomposable finite rank groups and their commensurability
A. R. Chekhlov^{} ^{} Tomsk State University, Tomsk, Russian Federation
Abstract:
A subgroup $H$ of an Abelian group $G$ is said to be fully inert in $G$ if the subgroup $H\cap\varphi H$ has a finite index in $\varphi H$ for any endomorphism $\varphi$ of the group $G$. Subgroups $H$ and $K$ of the group $G$ are said to be commensurable if the subgroup $K\cap H$ has a finite index in $H$ and in $K$. Some properties of fully inert and commensurable groups in the context of direct decompositions of the group and operations on subgroups are proved. For example, if a subgroup $H$ is commensurable with a subgroup $K$, then $H$ is commensurable with $H\cap K$ and with $H + K$; if a subgroup $H$ is commensurable with a subgroup $K$, then the subgroup $fH$ is commensurable with $fK$ for any homomorphism $f$. The main result of the paper is that every fully inert subgroup of a completely decomposable finite rank torsionfree group $G$ is commensurable with a fully invariant subgroup if and only if types of rank $1$ direct summands of the group $G$ are either equal or incomparable, and all rank $1$ direct summands of the group $G$ are not divisible by any prime number $p$.
Keywords:
factor group, fully invariant subgroup, commensurable subgroups, divisible hull, rank of the group.
DOI:
https://doi.org/10.17223/19988621/41/4
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UDC:
512.541 Received: 21.03.2016
Citation:
A. R. Chekhlov, “Fully inert subgroups of completely decomposable finite rank groups and their commensurability”, Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2016, no. 3(41), 42–50
Citation in format AMSBIB
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\by A.~R.~Chekhlov
\paper Fully inert subgroups of completely decomposable finite rank groups and their commensurability
\jour Vestn. Tomsk. Gos. Univ. Mat. Mekh.
\yr 2016
\issue 3(41)
\pages 4250
\mathnet{http://mi.mathnet.ru/vtgu526}
\crossref{https://doi.org/10.17223/19988621/41/4}
\elib{http://elibrary.ru/item.asp?id=26224725}
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This publication is cited in the following articles:

A. R. Chekhlov, “On Fully Inert Subgroups of Completely Decomposable Groups”, Math. Notes, 101:2 (2017), 365–373

A. R. Chekhlov, “On Strongly Invariant Subgroups of Abelian Groups”, Math. Notes, 102:1 (2017), 106–110

A. R. Chekhlov, “Intermediately fully invariant subgroups of abelian groups”, Siberian Math. J., 58:5 (2017), 907–914

U. Dardano, D. Dikranjan, S. Rinauro, “Inertial properties in groups”, Int. J. Group Theory, 7:3, 3 (2018), 17–62

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