Let H be a subgroup of G with index 2.
a. Prove that H is a normal subgroup of G.
b. Prove that g^(2) E H for all g E G.
There is a fairly well-known theorem which says that "if all the right cosets of a subgroup are left cosets of the subgroup then the subgroup is normal". So, notice that if are the left cosets and the right then (where that funny symbol just means union but the sets are pairwise disjoint). What now?
A more general theorem says that if then for all . To see this think about . What's it's order? what isb. Prove that g^(2) E H for all g E G.
For the first part:
for g E G
if g E H , then gH = H = Hg. Therefore H is a normal subgroup.
if g is not an element of H, G = eH U gH because there are only 2 cosets.
also G = He U Hg.
So G = H U Hg and G = H U gH (H = eH = He)
Therefore, gH = Hg. H is a normal subgroup.
I don't know how to prove part b. Sorry