Separability properties of nilpotent ℚ[𝑥]-powered groups II

Abstract In this paper, we study nilpotent <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="double-struck">Q</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:mrow> </m:math> \mathbb{Q}[x] -powered groups that satisfy the following property: for some set of primes 𝜔 in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="double-struck">Q</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:mrow> </m:math> \mathbb{Q}[x] , every <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ω</m:mi> <m:mo>′</m:mo> </m:msup> </m:math> \omega^{\prime} -isolated <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="double-struck">Q</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:mrow> </m:math> \mathbb{Q}[x] -subgroup in some family of its <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="double-struck">Q</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:mrow> </m:math> \mathbb{Q}[x] -subgroups is finite 𝜔-type separable.