Every subgroup of an abelian group is normal
Proof. Let N be the subgroup of the abelian group G. We want to show that for all g \in G, for all n \in N that gng^{-1} \in N. Since each subgroup of an abelian group is abelian, which have seen here, we know that gn = ng and therefore gn \in Ng. Taking the inverse on both sides, we get gng^{-1} \in N, which proves that every subgroup of the abelian group G is indeed normal.
