Universal Property of Abelianization of Group

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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.

$\blacksquare$