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I have a proof, but I wonder known another proofs. My proof:
Let $H\leq G$ be a subgroup of $G$. Let $H$ act on the coset space $(G/H)\setminus\{H\}$. By the orbit-stab.theorem and the assumption, you can easily see that all orbits of the coset space are singletons. If we define the action rule $h.gH=hgH$ , we get that $H$ is normal.

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There may be something of interest at math.stackexchange.com/questions/112107/…? In any event, I'm sure the question has been discussed on this site before. It may be worthwhile to search for it a bit, starting with the Related questions running down the right side of this page. – Gerry Myerson Nov 30 '12 at 21:58
    
Here is basically a duplicate. – JSchlather Nov 30 '12 at 22:09

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