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$\begingroup$

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 http://math.stackexchange.com/questions/112107/subgroup-of-maximal-order-is-normal? 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.2012-11-30
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    [Here](http://math.stackexchange.com/questions/164244/normal-subgroup-of-prime-index/164251#164251) is basically a duplicate.2012-11-30

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