Suppose that $A$ is a subset of a (large) finite cyclic group such that $|A|=5$ and $|A+A|=12$. Given that $g$ is a group element with $g+A\subseteq A+A$, can one conclude that $g\in A$? 



Does $g+A\subseteq A+A$ imply $g\in A$?
This is not the case. Here is a counter example.


Let $G =Z\_m, \ \ m>20$ (say). Let $A = \{0, 1, 3, 4, 6\}; g=2$: Note that $g+A=\{2, 3, 5, 6, 8\} \subseteq \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12\}=A+A$ and that $g \not\in A$, $|A|=5$, and $|A+A|=12$. Note that this example works for all large finite cyclic groups, as the reduction modulo $m$ does not matter.


