U+22B3 Contains As Normal Subgroup

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In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate 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 gin G} and n ∈ N . {displaystyle nin N.} The usual notation for this relation is N ◃ G . {displaystyle N riangleleft 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 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.