Geodesic
Fully inert subgroups of completely decomposable finite rank groups and their commensurability
Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 3 (2016), pp. 42-50

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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 torsion-free 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. 
A. R. Chekhlov. Fully inert subgroups of completely decomposable finite rank groups and their commensurability. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 3 (2016), pp. 42-50. http://geodesic.mathdoc.fr/item/VTGU_2016_3_a3/
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