It is proved that a suitable free Abelian group of finite rank is not absolutely closed in the class \( {\mathcal A} \) 2 of metabelian groups. A condition is specified under which a torsion-free Abelian group is not absolutely closed in \( {\mathcal A} \) 2. Also we gain insight into the question when the dominion in \( {\mathcal A} \) 2 of the additive group of rational numbers coincides with this subgroup.
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Translated from Algebra i Logika, Vol. 51, No. 5, pp. 608-622, September-October, 2012.
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Budkin, A.I. Dominions in abelian subgroups of metabelian groups. Algebra Logic 51, 404–414 (2012). https://doi.org/10.1007/s10469-012-9200-y
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DOI: https://doi.org/10.1007/s10469-012-9200-y