This paper proves a Freiman–Ruzsa-type theorem with polynomial bounds in arbitrary abelian groups with bounded torsion, thereby proving a conjecture of Marton. The main result states that if G is an abelian group of torsion m and A is a non-empty subset of G with |A + A| ≤ K|A|, then A can be covered by at most (2K)^{O(m³)} translates of a subgroup H of G with |H| ≤ |A|. The argument is a variant of that used in the case G = F₂ⁿ in a recent paper of the authors. The paper also includes a corollary related to the polynomial Bogolyubov conjecture and provides a polynomially effective inverse theorem for the U³(F_pⁿ)-norm. The proof involves entropy notions and entropic Ruzsa distance, and uses a multidistance chain rule to establish the main result. The paper concludes with a detailed proof of the main theorem and its implications.This paper proves a Freiman–Ruzsa-type theorem with polynomial bounds in arbitrary abelian groups with bounded torsion, thereby proving a conjecture of Marton. The main result states that if G is an abelian group of torsion m and A is a non-empty subset of G with |A + A| ≤ K|A|, then A can be covered by at most (2K)^{O(m³)} translates of a subgroup H of G with |H| ≤ |A|. The argument is a variant of that used in the case G = F₂ⁿ in a recent paper of the authors. The paper also includes a corollary related to the polynomial Bogolyubov conjecture and provides a polynomially effective inverse theorem for the U³(F_pⁿ)-norm. The proof involves entropy notions and entropic Ruzsa distance, and uses a multidistance chain rule to establish the main result. The paper concludes with a detailed proof of the main theorem and its implications.