In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup H of a group G is normal in G if and only if aH = Ha for all a in G (see coset). Normal subgroups (and only normal subgroups) can be used to construct quotient groups from a given group.
Évariste Galois was the first to realize the importance of the existence of normal subgroups.
Read more about Normal Subgroup: Definitions, Examples, Properties, Normal Subgroups and Homomorphisms
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—Elizabeth Cady Stanton (1815–1902)