Let H have index 2 in a group G. Let g be any element of G. If g∈H,
then gH=H=Hg. If g is not in H, then, since there are exactly two left cosets
of H in G and g is not in H, they must be H and gH. Since left cosets are 
disjoint, we know gH=G−H. But right cosets are also disjoint, so Hg=G−H. 
Hence Hg=G−H=gH. Thus gH=Hg for all g∈G, so H is normal.