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We prove a Freiman-Ruzsa-type theorem valid in an arbitrary nilpotent group.Specifically, we show that a K-approximate subgroup A of an s-step nilpotentgroup G is contained in a coset nilprogression of rank at most f(K) andcardinality at most exp(g(K))|A|, with f and g polynomials depending only onthe step s of G. To motivate this, we give a direct proof of Breuillard andGreen's analogous result for torsion-free nilpotent groups, avoiding the use ofMal'cev's embedding theorem.

Freiman's theorem in an arbitrary nilpotent group