Separability Properties of Nilpotent $\mathbb{Q}[x]$-Powered Groups II
Stephen Majewicz, Marcoz Zyman
TL;DR
This paper extends separability and residual results from classical nilpotent groups to nilpotent $\mathbb{Q}[x]$-powered groups by introducing and analyzing the $\mathcal{G}_{\omega}$ framework. It proves that for abelian $\mathbb{Q}[x]$-groups, $\mathcal{G}_{\omega}$ is equivalent to Condition ($B_{\omega}$), and shows that $\omega$-restricted nilpotent $\mathbb{Q}[x]$-powered groups have $\mathcal{G}_{\omega}$ with respect to their normal subgroups; additionally, torsion-free cases imply the inheritance of $\mathcal{G}_{\omega}$ across central factors. The results unify separability, residual finiteness, and torsion decomposition in this binomial-ring setting and establish a structural pathway for analyzing isolated subgroups and quotients, with an open problem proposed for broader applicability across classes. These insights advance understanding of subgroup separability in algebraic structures defined over $\mathbb{Q}[x]$.
Abstract
In this paper, we study nilpotent $\mathbb{Q}$$[x]$-powered groups that satisfy the following property: For some set of primes $ω$ in $\mathbb{Q}$$[x]$, every $ω'$-isolated $\mathbb{Q}$$[x]$-subgroup in some family of its $\mathbb{Q}$$[x]$-subgroups is finite $ω$-type separable.