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In abstract algebra, a normal subgroup is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N {\displaystyle N} of the group G {\displaystyle G} is normal in G {\displaystyle G} if and only if g n g − 1 ∈ N {\displaystyle gng^{-1}\in N} for all g ∈ G {\displaystyle g\in G} and n ∈ N . {\displaystyle n\in N.} The usual notation for this relation is N ◃ G . {\displaystyle N\triangleleft G.}
Normal subgroups are important because they can be used to construct quotient groups of the given group. Furthermore, the normal subgroups of G {\displaystyle G} are precisely the kernels of group homomorphisms with domain G , {\displaystyle G,} which means that they can be used to internally classify those homomorphisms.
Évariste Galois was the first to realize the importance of the existence of normal subgroups.