Assume $S$ is a subgroup of group $G$
How to prove this:
Two elements are in the same coset of $S$ iff their difference is in $S$
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HINT: Suppose that $x,y\in G$, and $x-y\in S$. Let $s=x-y$. Then $x=s+y\in S+y$. In other words, the coset of $S$ containing $x$ is the coset $S+y$. To complete the proof you need only show that $y\in S+y$ as well.
Note that all I did was translate the hypothesis into specifics and make one pretty obvious algebraic manipulation, from $x-y\in S$ to $x\in S+y$ to get a coset of $S$ into the picture. Getting to this point requires should be (or soon become) almost automatic.