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For the class of solvable groups of homeomorphisms of the line preservingorientation and containing a freely acting element, we establish themetabelianity of the quotient group $G/H_G$, where the elements of the normalsubgroup $H_G$ are stabilizers of the minimal set. This fact is an importantelement in the classification theorem, used, in particular, in the study of theThompson's group $F$.

On the metabelian property of quotient groups of solvable groups of orientation-preserving homeomorphisms of the line