Universal Property of Quotient Group
Theorem
Let $G$ and $H$ be groups.
Let $N \trianglelefteq G$ be an normal subgroup.
Let $\pi: G \to G / N$ be the quotient epimorphism.
Let $f: G \to H$ be a group homomorphism with $N \subseteq \ker f$.
Then there exists a unique group homomorphism $\overline f: G / N \to H$ such that $f = \overline f \circ \pi$.
$\xymatrix{ G \ar[d]^\pi \ar[r]^{\forall f} & H \\ G/N \ar[ru]_{\exists ! \bar f} }$
Proof
Existence
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Let $\sim$ denote (left) congruence modulo $N$.
From Congruence Modulo Subgroup is Equivalence Relation, $\sim$ is an equivalence relation on $X$.
For all $g \in G$, let $\eqclass g \sim$ denote the equivalence class of $g$ under $\sim$.
Note that Group Homomorphism is Invariant under Congruence Modulo Kernel.
In particular, since $N \subseteq \ker f$, we have that $f$ is $\sim$-invariant.
Hence, by Universal Property of Quotient Set, there exists a unique such mapping $\overline f: G / \sim{} \to H$, which is given by:
- $\forall g \in G: \map {\overline f} {\eqclass g \sim} = \map f g$
From Universal Property of Quotient Set, $f$ is well-defined.
By definition of (left) coset space, $G / N = G / \sim$.
By Left Cosets are Equal iff Product with Inverse in Subgroup:
- for all $g \in G$ the equivalence class $\eqclass g \sim$ is the (left) coset of $N$ containing $g$, which is $g N$.
Thus $\overline f$ is given by:
- $\forall g \in G: \map {\overline f} {g N} = \map f g$
We have that $\overline f$ is a group homomorphism.
Indeed, for all $x, y \in G$ we have:
| \(\ds \map {\overline f} {\paren {x N} \paren {y N} }\) | \(=\) | \(\ds \map {\overline f} {\paren {x y} N}\) | Definition of Quotient Group | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map f {x y}\) | Definition of $\overline f$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map f x \map f y\) | Definition of Group Homomorphism | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map {\overline f} {x N} \map {\overline f} {y N}\) | Definition of $\overline f$ |
Thus we have shown the existence of such a group homomorphism.
$\Box$
Uniqueness
By Universal Property of Quotient Set, there exists a unique such mapping $\overline f$.
A fortiori, there exists at most one such group homomorphism.
$\blacksquare$