Let $G$ a finite group and $H$ a sugroup of $G$. Show that $H$ is normal if and only if all the double cosets $HgH$ for every $g\in G$ have equal number of elements.
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I only need to show that if the double classes have the same number of elements then H is normal. Thanks – Roiner Segura Cubero Oct 01 '15 at 18:07
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If $h\in H$ then the double coset $HhH$ is $H$, so suppose that all double cosets have size $|H|$.
Take $g\in G$. The right coset $Hg$ is contained in the double coset $HgH$ (just take the right hand multiplier to be $e$. But $|Hg|=|H|$ so we must have $Hg=HgH$. Similarly $gH=HgH$ whence $Hg=gH$ and we are done.
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