On dominions of the rationals in nilpotent groups (Q1617976)

A quasivariety of groups $\mathcal{M}$ is a class of groups which are defined by a set of quasiidentities, i.e. expressions of the form \[ u_1 = u_2 = \dots = u_k = e \implies v = e \] where $u_i$, $v$ are some words in arbitrary elements of the group. For example, torsion-free groups are defined by the quasiidentities \[ x^p = e \implies x = e \] for all primes $p$. The paper under reviews mostly considers quasivarieties of torsion-free nilpotent groups of class $\leq 3$. \par Given an arbitrary quasivariety of groups $\mathcal{M}$, a group $G \in \mathcal{M}$ and a subgroup $H \le G$, we define the dominion $\mathrm{dom}^\mathcal{M}_G(H)$ of $H$ in $G$ as follows. An element $x \in G$ lies in $\mathrm{dom}^\mathcal{M}_G(H)$ if and only if for all group morphisms $f, g: G \to M$ with $M \in \mathcal{M}$ such that $f(h) = g(h)$ for all $h \in H$, then automatically $f(x) = g(x)$. The dominion $\mathrm{dom}^\mathcal{M}_G(H)$ is hence a subgroup of $G$ containing $H$ and were introduced for the first time in [\textit{J. R. Isbell}, in: Proc. Conf. Categor. Algebra, La Jolla 1965, 232--246 (1966; Zbl 0194.01601)] and [\textit{J. M. Howie} and \textit{J. R. Isbell}, J. Algebra 6, 7--21 (1967; Zbl 0211.33303)] for their importance in studying epimorphisms. \par In the paper under review, the author shows that the additive group of rational numbers $\mathbb{Q}$ is $2$-closed in every quasivariety $\mathcal{M}$ of torsion-free nilpotent groups of class $\leq 3$. This means that for every group $G = \langle \mathbb{Q}, a_1, a_2 \rangle$ which contains $\mathbb{Q}$, the dominion $\mathrm{dom}^\mathcal{M}_G(\mathbb{Q}) = \mathbb{Q}$. The proof of this result is technical, making many computations in presentations of groups and generalizes previous work of the author.