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 $g\in A$?

2$\begingroup$ The condition "large" is weird: indeed, if there is a counterexample in a cyclic group $C_m$, then we get the same counterexample in $C_{m^n}$ for all $n$. Hence, either you want to omit it (but then you maybe know a counterexample), or you want to replace "large" with "with no small prime divisor". Or make some further assumption, e.g., that $A$ generates $C_m$. $\endgroup$– YCorApr 6 '20 at 10:35

6$\begingroup$ Why $5$, Seva? why $12$? What's the real question here? $\endgroup$– Gerry MyersonApr 6 '20 at 12:24

1$\begingroup$ Good question, Gerry; I actually expected someone to ask it!. If the answer were positive, I would be able to exclude one of the countless cases emerging in some large project. $\endgroup$– SevaApr 6 '20 at 12:27
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.