Hi next one? Any ideas here?
Let G be a group and a subgroup such that |G/H| = 2. Show that H is normal in G.
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.