Isolators in soluble groups of finite rank
A group G has the isolator property if for every subgroup H of G the set \(\sqrt{H}=\{g\in G:\quad g^ n\in H\quad for\quad some\quad n\geq 1\}\) is a subgroup and G has the strong isolator property if in addition \(| \sqrt{H}:H|\) is finite whenever H/\(\cap_{g\in G}H^ g\) is finitely generated. For any property \({\mathcal P}\) of groups we say that G almost has \({\mathcal P}\) if G has a subgroup of finite index satisfying \({\mathcal P}.\) The main results are Theorem A. A finitely generated soluble group is polycyclic if and only if it almost has the strong isolator property. - Theorem B. A torsion-free soluble group of finite rank almost has the isolator property. If G is a finitely generated nilpotent by polycyclic by finite group with the isolator property, then G has finite rank. - Theorem C. A finitely generated linear group almost has the isolator property if and only if it is almost soluble of finite rank.
torsion-free soluble group of finite rank
finitely generated nilpotent by polycyclic by finite group
scientific article; zbMATH DE number 3867573