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 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 | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map f {x^{-1} \paren {x y^{-1} } y}\) | 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 Derived 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 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.
$\blacksquare$