Freiman's theorem in an arbitrary abelian group

A famous result of Freiman describes the structure of finite sets A subset of Z with small doubling property. If vertical bar A + A vertical bar <= K vertical bar A vertical bar, then A is contained within a multidimensional arithmetic progression of dimension d(K) and size f(K) vertical bar A vertical bar. Here we prove an analogous statement valid for subsets of an arbitrary abelian group.