Normal subgroup

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In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup)^[1] is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup ${\displaystyle N}$ of the group ${\displaystyle G}$ is normal in ${\displaystyle G}$ if and only if ${\displaystyle gng^{-1}\in N}$ for all ${\displaystyle g\in G}$ and ${\displaystyle n\in N}$. The usual notation for this relation is ${\displaystyle N\triangleleft G}$.

Normal subgroups are important because they (and only they) can be used to construct quotient groups of the given group. Furthermore, the normal subgroups of ${\displaystyle G}$ are precisely the kernels of group homomorphisms with domain ${\displaystyle G}$, 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.^[2]