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On dominions of the rationals in nilpotent groups
A. I. Budkin Altai State University, Barnaul, Russia
Abstract:
The dominion of a subgroup $H$ of a group $G$ in a class $M$ is the set of all $a\in G$ that have the same images under every pair of homomorphisms, coinciding on $H$ from $G$ to a group in $M$. A group $H$ is $n$-closed in $M$ if for every group $G=\operatorname{gr}(H,a_1,\dots,a_n)$ in $M$ that includes $H$ and is generated modulo $H$ by some $n$ elements, the dominion of $H$ in $G$ (in $M$) is equal to $H$. We prove that the additive group of the rationals is $2$-closed in every quasivariety of torsion-free nilpotent groups of class at most $3$.
Keywords:
quasivariety, nilpotent group, additive group of the rationals, dominion, $2$-closed group.
Received: 18.11.2017
Citation:
A. I. Budkin, “On dominions of the rationals in nilpotent groups”, Sibirsk. Mat. Zh., 59:4 (2018), 759–772; Siberian Math. J., 59:4 (2018), 598–609
Linking options:
https://www.mathnet.ru/eng/smj3008 https://www.mathnet.ru/eng/smj/v59/i4/p759
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Abstract page: | 253 | Full-text PDF : | 50 | References: | 49 | First page: | 2 |
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