# Universal Property of Abelianization of Group

## Theorem

Let $G$ be a group.

Let $G^{\operatorname {ab} }$ be its abelianization.

Let $\pi : G \to G^{\operatorname {ab} }$ be the quotient group epimorphism.

Let $H$ be an abelian group.

Let $f: G \to H$ be a group homomorphism.

Then there exists a unique group homomorphism $g : G^{\operatorname {ab}} \to H$ such that $g \circ \pi = f$:

$\xymatrix { G \ar[d]_\pi \ar[r]^{\forall f} & H\\ G^{\operatorname {ab} } \ar[ru]_{\exists ! g} }$

## Proof 1

Let $e_G$ be the identity element of $G$.

Let $e_H$ be the identity element of $H$.

Then:

 $\ds \forall x, y \in G: \,$ $\ds \map f {\sqbrk {x, y} }$ $=$ $\ds \map f {x^{-1} y^{-1} x y}$ Definition of Commutator $\ds$ $=$ $\ds \map f {x^{-1} } \paren {\map f {y^{-1} } \map f x} \map f y$ Definition of Group Homomorphism $\ds$ $=$ $\ds \map f {x^{-1} } \paren {\map f x \map f {y^{-1} } } \map f y$ Definition of Commutative Operation $\ds$ $=$ $\ds \map f {x^{-1} \paren {x y^{-1} } y}$ Definition of Group Homomorphism $\ds$ $=$ $\ds \map f {\paren {x^{-1} x} \paren {y^{-1} y} }$ Associativity on Four Elements $\ds$ $=$ $\ds \map f {e_G}$ $\ds$ $=$ $\ds e_H$ Group Homomorphism Preserves Identity $\ds \leadsto \ \$ $\ds \forall x, y \in G: \,$ $\ds \sqbrk {x, y}$ $\in$ $\ds \ker f$ Definition of Kernel of Group Homomorphism $\ds \leadsto \ \$ $\ds \sqbrk {G, G}$ $\subseteq$ $\ds \ker f$ Definition of Commutator Subgroup

The result follows from the definition of abelianization and Universal Property of Quotient Group.

$\blacksquare$

## Proof 2

Let $e_H$ be the identity element of $H$.

Then:

 $\ds \forall x, y \in G: \,$ $\ds \map f {\sqbrk {x, y} }$ $=$ $\ds \map f {x^{-1} y^{-1} x y}$ Definition of Commutator $\ds$ $=$ $\ds \map f {x^{-1} } \paren {\map f {y^{-1} } \map f x} \map f y$ Definition of Group Homomorphism $\ds$ $=$ $\ds \map f {x^{-1} } \paren {\map f x \map f {y^{-1} } } \map f y$ Definition of Commutative Operation $\ds$ $=$ $\ds \paren {\map f x}^{-1} \paren {\map f x \paren {\map f y}^{-1} } \map f y$ Group Homomorphism Preserves Inverses $\ds$ $=$ $\ds \paren {\paren {\map f x}^{-1} \map f x} \paren {\paren {\map f y}^{-1} \map f y}$ Associativity on Four Elements $\ds$ $=$ $\ds e_H$ $\ds \leadsto \ \$ $\ds \forall x, y \in G: \,$ $\ds \sqbrk {x, y}$ $\in$ $\ds \ker f$ Definition of Kernel of Group Homomorphism $\ds \leadsto \ \$ $\ds \sqbrk {G, G}$ $\subseteq$ $\ds \ker f$ Definition of Commutator Subgroup

The result follows from the definition of abelianization and Universal Property of Quotient Group.

$\blacksquare$