Universal Property of Abelianization of Group/Proof 2

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

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 of Group Elements

\(\ds \)

\(=\)

\(\ds \map f {x^{-1} } \paren {\map f {y^{-1} } \map f x} \map f y\)

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 Derived Subgroup

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

$\blacksquare$