Editing Normal subgroup (section)

== Definitions ==
A [[subgroup]] <math>N</math> of a group <math>G</math> is called a '''normal subgroup''' of <math>G</math> if it is invariant under [[inner automorphism|conjugation]]; that is, the conjugation of an element of <math>N</math> by an element of <math>G</math> is always in <math>N.</math>{{sfn|Dummit|Foote|2004}}  The usual notation for this relation is <math>N \triangleleft G.</math>

===Equivalent conditions===
For any subgroup <math>N</math> of <math>G,</math> the following conditions are [[Logical equivalence|equivalent]] to <math>N</math> being a normal subgroup of <math>G.</math> Therefore, any one of them may be taken as the definition.

* The image of conjugation of <math>N</math> by any element of <math>G</math> is a subset of <math>N,</math>{{sfn|Hungerford|2003|p=41}} i.e., <math>gNg^{-1}\subseteq N</math> for all <math>g\in G</math>.
* The image of conjugation of <math>N</math> by any element of <math>G</math> is equal to <math>N,</math>{{sfn|Hungerford|2003|p=41}} i.e., <math>gNg^{-1}= N</math> for all <math>g\in G</math>.
* For all <math>g \in G,</math> the left and right cosets <math>gN</math> and <math>Ng</math> are equal.{{sfn|Hungerford|2003|p=41}}
* The sets of left and right [[coset]]s of <math>N</math> in <math>G</math> coincide.{{sfn|Hungerford|2003|p=41}}
* Multiplication in <math>G</math> preserves the equivalence relation "is in the same left coset as". That is, for every <math>g,g',h,h'\in G</math> satisfying <math>g N = g' N</math> and <math>h N = h' N</math>, we have <math>(g h) N = (g' h') N.</math>
* There exists a group on the set of left cosets of <math>N</math> where multiplication of any two left cosets <math>gN</math> and <math>hN</math> yields the left coset <math>(gh)N</math>. (This group is called the ''quotient group'' of <math>G</math> ''modulo'' <math>N</math>, denoted <math>G/N</math>.)
* <math>N</math> is a [[Union (set theory)|union]] of [[conjugacy class]]es of <math>G.</math>{{sfn|Cantrell|2000|p=160}}
* <math>N</math> is preserved by the [[inner automorphism]]s of <math>G.</math>{{sfn|Fraleigh|2003|p=141}}
* There is some [[group homomorphism]] <math>G \to H</math> whose [[Kernel (algebra)|kernel]] is <math>N.</math>{{sfn|Cantrell|2000|p=160}}
* There exists a group homomorphism <math>\phi:G \to H</math> whose [[Fiber (mathematics)|fibers]] form a group where the identity element is <math>N</math> and multiplication of any two fibers <math>\phi^{-1}(h_1)</math> and <math>\phi^{-1}(h_2)</math> yields the fiber <math>\phi^{-1}(h_1 h_2)</math>. (This group is the same group <math>G/N</math> mentioned above.)
* There is some [[congruence relation]] on <math>G</math> for which the [[equivalence class]] of the [[identity element]] is <math>N</math>.
* For all <math>n\in N</math> and <math>g\in G,</math> the [[commutator]] <math>[n,g] = n^{-1} g^{-1} n g</math> is in <math>N.</math>{{cn|date=March 2019}}
* Any two elements commute modulo the normal subgroup membership relation.  That is, for all <math>g, h \in G,</math> <math>g h \in N</math> if and only if <math>h g \in N.</math>{{cn|date=October 2020}}