Hi next one? Any ideas here?

Let G be a group and $\displaystyle H \subset G$ a subgroup such that |G/H| = 2. Show that H is normal in G.

thnx :)

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- Feb 22nd 2008, 06:21 AMjoanne_qNormality
Hi next one? Any ideas here?

Let G be a group and $\displaystyle H \subset G$ a subgroup such that |G/H| = 2. Show that H is normal in G.

thnx :) - Feb 22nd 2008, 07:35 AMThePerfectHacker
We want to show that aH = Ha for any a in G to prove normality. If a in H then aH = H = Ha and proof is complete. If a not in H then aH is not H, similarly Ha is not H. But since there are only two cosets and both of them are not H it means they are the same so aH = Ha.