Editing Normal subgroup (section)

{{Short description|Subgroup invariant under conjugation}}
{{Redirect-distinguish|Invariant subgroup|Fully invariant subgroup}}
{{Group theory sidebar|Basics}}
In [[abstract algebra]], a '''normal subgroup''' (also known as an '''invariant subgroup''' or '''self-conjugate subgroup'''){{sfn|Bradley|2010|p=12}} is a [[subgroup]] that is [[Invariant (mathematics)|invariant]] under [[inner automorphism|conjugation]] by members of the [[Group (mathematics)|group]] of which it is a part. In other words, a subgroup <math>N</math> of the group <math>G</math> is normal in <math>G</math> if and only if <math>gng^{-1} \in N</math> for all <math>g \in G</math> and <math>n \in N.</math> The usual notation for this relation is <math>N \triangleleft G.</math>

Normal subgroups are important because they (and only they) can be used to construct [[quotient group]]s of the given group.  Furthermore, the normal subgroups of <math>G</math> are precisely the [[Kernel of a homomorphism|kernels]] of [[Group homomorphism|group homomorphisms]] with [[domain of a function|domain]] <math>G,</math> 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.{{sfn|Cantrell|2000|p=160}}