Product of normal subgroups is normal

If N and M are normal in G then NM is a normal subgroup of G

Product of normal subgroups is normal

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

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

Then $NM$ is a normal subgroup of $G$ . Moreover, $NM=MN$ .

Context. Products like $NM$ appear in building larger normal subgroups from smaller ones (e.g. in series and extensions). The equality $NM=MN$ is a typical “normality makes products commute setwise” phenomenon.