Freiman isomorphism

Definition
Suppose $$G, H$$ are abelian groups, and $$S \subseteq G, T \subseteq H$$ are subsets. A Freiman isomorphism of rank $$r$$ between $$S$$ and $$T$$ is a bijection $$\varphi:S \to T$$ such that:


 * For $$s_1,s_2,\dots,s_j \in S$$, with $$j \le r$$, (note: the $$s_i$$ need not be distinct), we have:

$$s_1 + s_2 + \dots + s_j \in S \iff \varphi(s_1) + \varphi(s_2) + \dots + \varphi(s_j) \in T$$


 * If either side is well-defined, we have:

$$\varphi(s_1 + s_2 + \dots + s_j) = \varphi(s_1) + \varphi(s_2) + \dots + \varphi(s_j)$$.

Particular cases

 * Any bijection is a Freiman isomorphism of rank one.
 * A bijection is a Freiman isomorphism of rank two if and only if whenever the sum of two elements in one set lies in the set, the same happens for the other set, and these sums correspond.

Facts

 * Every finite subset of an abelian group is Freiman-isomorphic to a subset of a finite abelian group
 * The noncommutative analogue fails to hold.