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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.

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  • $\begingroup$ Thanks. What about the converse? is this right? : $x+s_1 - (x+s_2) = s_1 - s_2 \in S$ ? $\endgroup$ – Mahdi Khosravi Oct 11 '13 at 13:45
  • $\begingroup$ @Mahdi: You’re welcome. Yes, that gives you the other direction. $\endgroup$ – Brian M. Scott Oct 11 '13 at 18:56

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