Let $(G,+)$ be an abelian group and $A$, $B$ and $C$ be finite subsets of $G$ with $A+B=C$. One may conclude that $A\subset C-B$. However, $A$ need not be equal to $C-B$. What is a necessary and sufficient condition to have $A=C-B$?



Sumsets with the property "$A+B=C$ implies $A=C-B$"
Regardless of whether $A,\ B$, and $G$ are finite or infinite, the necessary and sufficient condition is that $B$ lies in a coset of the *period* (or *stabilizer*) of $A$, defined to be the subgroup of all those group elements $g$ with $A+g=A$.


To see this, notice that $A\subseteq (A+B)-B$ holds true in a trivial way, so that your condition reduces to $(A+B)-B\subseteq A$, which is thus equivalent to $A+(b\_1-b\_2)=A$ for any $b\_1,b\_2\in B$.


