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Let $A$ be a subset of $G$, where $G$ is a finite abelian group of torsion$r$. It was conjectured by Ruzsa that if $|A+A|\leq K|A|$, then $A$ iscontained in a coset of $G$ of size at most $r^{CK}|A|$ for some constant $C$.The case $r=2$ received considerable attention in a sequence of papers, and wasresolved by Green and Tao. Recently, Even-Zohar and Lovett settled the casewhen $r$ is a prime. In this paper, we confirm the conjecture when $r$ is apower of prime. In particular, the bound we obtain is tight.

Small doublings in abelian groups of prime power torsion