Editing Commutator subgroup (section)

=== Abelianization ===
Given a group <math>G</math>, a [[quotient group]] <math>G/N</math> is abelian if and only if <math>[G, G]\subseteq N</math>.

The quotient <math>G/[G, G]</math> is an abelian group called the '''abelianization''' of <math>G</math> or <math>G</math> '''made abelian'''.<ref>{{harvtxt|Fraleigh|1976|p=108}}</ref> It is usually denoted by <math>G^{\operatorname{ab}}</math> or <math>G_{\operatorname{ab}}</math>.

There is a useful categorical interpretation of the map <math>\varphi: G \rightarrow G^{\operatorname{ab}}</math>.  Namely <math>\varphi</math> is universal for homomorphisms from <math>G</math> to an abelian group <math>H</math>: for any abelian group <math>H</math> and homomorphism of groups <math>f: G \to H</math> there exists a unique homomorphism <math>F: G^{\operatorname{ab}}\to H</math> such that <math>f = F \circ \varphi</math>.  As usual for objects defined by universal mapping properties, this shows the uniqueness of the abelianization <math>G^{\operatorname{ab}}</math> up to canonical isomorphism, whereas the explicit construction <math>G\to G/[G, G]</math> shows existence.

The abelianization functor is the [[adjoint functors|left adjoint]] of the inclusion functor from the [[category of abelian groups]] to the category of groups. The existence of the abelianization functor '''Grp''' → '''Ab''' makes the category '''Ab''' a [[reflective subcategory]] of the category of groups, defined as a full subcategory whose inclusion functor has a left adjoint. 

Another important interpretation of <math>G^{\operatorname{ab}}</math> is as <math>H_1(G, \mathbb{Z})</math>, the first [[group homology|homology group]] of <math>G</math> with integral coefficients.