The rectifiability threshold in abelian groups

For any abelian group G and integer t ≥ 2 we determine precisely the smallest possible size of a non-t-rectifiable subset of G. Specifically, assuming that G is not torsion-free, denote by p the smallest order of a non-zero element of G. We show that if a subset S ⊆ G satisfies |S| ≤ ⌌log _t p⌍, then S is t-isomorphic (in the sense of Freiman) to a set of integers; on the other hand, we present an example of a subset S ⊆ G with |S| = ⌌log _t p⌍ + 1 which is not t-isomorphic to a set of integers.