# Separability Properties of Nilpotent $\mathbb{Q}[x]$-Powered Groups

## TL;DR

This paper investigates conjugacy and subgroup separability in nilpotent groups extended over the polynomial ring f[x], generalizing known theorems and exploring structural properties of these algebraic objects.

## Contribution

It generalizes Baumslag's theorem to finitely f[x]-generated, torsion-free nilpotent f[x]-powered groups, establishing new intersection properties for f[x]-isolated subgroups.

## Key findings

- Generalization of Baumslag's theorem to f[x]-powered groups
- Establishment of intersection properties for f[x]-isolated subgroups
- Extension of techniques from nilpotent groups to f[x]-powered groups

## Abstract

In this paper we study conjugacy and subgroup separability properties in the class of nilpotent $\mathbb{Q}[x]$-powered groups. Many of the techniques used to study these properties in the context of ordinary nilpotent groups carry over naturally to this more general class. Among other results, we offer a generalization of a theorem due to G. Baumslag. The generalized version states that if $G$ is a finitely $\mathbb{Q}[x]$-generated $\mathbb{Q}[x]$-torsion-free nilpotent $\mathbb{Q}[x]$-powered group and $H$ is a $\mathbb{Q}[x]$-isolated subgroup of $G,$ then for any prime $\pi \in \mathbb{Q}[x]$, $\bigcap_{i = 1}^{\infty} G^{{\pi}^{i}}H = H.$