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If N and M are normal in G then NM is a normal subgroup of G
Proposition (Product of normal subgroups). Let be a group and let be normal subgroups. Define
Then is a normal subgroup of . Moreover, .
Context. Products like appear in building larger normal subgroups from smaller ones (e.g. in series and extensions). The equality is a typical "normality makes products commute setwise" phenomenon.
A group is a set together with a binary operation such that:
Equivalently, a group is a monoid in which every element is invertible. Much of group theory studies subgroups and structure-preserving maps called group homomorphisms.
Let be a group and let be a subgroup. The subgroup is normal in (written ) if for every ,
Normality says that is stable under the conjugation action of on itself. Equivalently, is normal iff every left coset of equals the corresponding right coset, and this is exactly the hypothesis needed to form the quotient group .