Shachar Lovett University of California, San Diego (UCSD) Let GG be a finite abelian group of torsion rr and let AA be a subset of GG. The Freiman-Ruzsa theorem asserts that if |A+A|≤K|A||A+A|≤K|A| then AA is contained in a coset of a subgroup of size at most K2rK4|A|K2rK4|A|. Ruzsa conjectured that the bound can be improved to rcK|A|rcK|A| for some absolute constant c≥2c≥2. This conjecture was verified for r=2r=2 in a sequence of recent works. In this work, we establish the same conjecture for any prime torsion. Joint work with Chaim Even-Zohar.