Product of normal subgroups is normal

Proposition (Product of normal subgroups). Let $G$ be a group and let $N,M\trianglelefteq G$ be normal subgroups. Define

NM=\{nm:n\in N,\ m\in M\}.

Then $NM=MN$ , and this set is a normal subgroup of $G$ .

Remarks

Normality gives $nm=m(m^{-1}nm)\in MN$ , so $NM\subseteq MN$ ; symmetry gives equality. Conjugation by any $g\in G$ preserves both $N$ and $M$ , hence preserves $NM$ .