Prove that $S$ is a coset of some subgroup of $G$ iff $S+S-S=S.$ I'm a student self-learning abstract-algebra but encountered this problem. The full problem is here:

Suppose $S$ is a nonempty subset of an additive abelian group $G$. Prove that $S$ is a coset of some subgroup of $G$ iff $S+S-S=\{a+b-c:a,b,c \in S\}$ is equal to $S$.

My sketch of proof is like this:
($\Rightarrow$) I can prove this direction.
($\Leftarrow$) But for this direction, I can't figure it out. I tried two ways, the first is that suppose $x\in G$, want to show there exists $s \in S$ such that $s=x+h$, for some $h \in H$, then $x+h=a+b-c$, but I think I cannot find the relation between the coset and the subgroup $H$ through this equation.
The second approach is that since $G$ is additive abelian, then $H$ is a normal subgroup of $G$. I can also show that $S$ is a normal subgroup of $G$, but it seems like there isn't any relation between the coset $S$ and $H$, either.
Any discussions and help are really appreciated. Thanks!
 A: Since $S$ is non-empty, pick any $t\in S$. I claim that $S-t:=\{s-t:s\in S\}$ is a group, which will show the desired result. Clearly $0\in S-t$. Thus we need only show that, for any $x,y\in S-t$, we have $x-y\in S-t$. But if $x,y\in S-t$ then $t+x,t+y\in S$. Hence, since $S+S-S=S$, also $(t+x)+t-(t+y)\in S$, ie $t+x-y\in S$, ie $x-y\in S-t$, as desired, so we are done.
A: This is @AtticusStonestrom's proof in more detail.
Since $\varnothing\neq S$, let $t\in S$.
It suffices to show that $H:=S-t=\{ s-t\mid s\in S\}$ is a subgroup of $G$, since then $S=H+t$ is a coset of a subgroup of $G$.
We use the one-step subgroup test:

*

*We have $0=t-t\in H$. Thus $H$ is nonempty.


*By definition, $H\subseteq G$.


*Let $x,y\in H$. Then $x+t, y+t\in S$ by definition of $H$. Now $$z:=(x+t)+t-(y+t)\in S$$ by hypothesis (because we can let $a=x+t, b=t, c=y+t$). But $G$ is abelian and so $$\begin{align}
z&=x+t+t-y-t\\
&=(x-y)+t,
\end{align}$$ which implies $x-y=z-t\in H$ as $z\in S$.
Thus $H\le G$.
Therefore, $S$ is a coset of a subgroup of $G$.
