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>β’</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> Q[x] -powered groups that satisfy the following property: for some set of primes π in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="double-struck">Q</m:mi> <m:mo>β’</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> Q[x] , every <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>Ο</m:mi> <m:mo>β²</m:mo> </m:msup> </m:math> Ο^{} -isolated <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="double-struck">Q</m:mi> <m:mo>β’</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> 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>β’</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> Q[x] -subgroups is finite π-type separable.
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Majewicz et al. (2024) studied this question.
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