Suppose $G$ is a torsion-free abelian group (written additively) and $A$ and $B$ are two nonempty finite subsets (not subgroups) of $G$. Is it true that there is an element of $G$ which may be expressed as a sum $a + b$ where $a$ is in $A$ and $b$ is in $B$ such that this representation is unique?
I'm trying an induction proof but I'm running into difficulties. I can prove a simple case when one of the sets has $1$ or $2$ elements. The hypothesis that $G$ be torsion-free is necessary because otherwise there is an element $x$ in $G$ of order $n>1$ and one may take $A = \{0,x\}$ and $B = \{0,x,2x,...,(n-1)x\}$ in which case $0 + kx = x + (k-1)x$ for all $k$ so any sum is not unique.
Hint: You can restrict yourself to the finitely generated $G$. The finitely generated torsion-free abelian groups have a fairly simple structure.