= Finitely generated soluble torsion groups are finite
A finitely generated <abelian group> whose elements all have finite order is finite. Induct on derived length for a finitely generated <soluble group> with this property. Its <abelianization> is finite, so its <commutator subgroup> has finite index and is finitely generated by <Schreier's lemma>. That subgroup is soluble of smaller derived length and is still a <torsion group>, hence finite by induction. The ambient group is a finite extension of it and is finite.
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