Prove that if H has index 2 in G, then H is normal in G.
With the definition of index being the number of disticnts cosets of H in G.
On first glance I don't have much on this, I am leaning toward doing it by contradiction, because I don't see any direct correlation between the two topics.
Question: Prove that if H has index 2 in G, then H is normal in G.
Answer:
Let G be a group and H be the subgroup of G.
In order of H to be normal in G, h $ \in $ H and g $ \in $ G then, gh $ g^(-1) $ $ \in $ H
So, if H = { H , ah }, and if a $ \in $ H, then aH = H = Ha.
If x is not $ \in $ H, then aH $ \in $ G but not H and Ha $ \in $ G too but not in H.
