----- > [!theorem] Theorem. ([[characterization of quotienting a group]]) > Let $G$ be a [[group]] and $\sim$ an [[equivalence relation]] on $G$. The following are equivalent: > 1. It is possible to endow $G/{\sim}$ with [[group]] structure such that $\pi_{\sim}$ is a [[group homomorphism]]; > 2. $\sim$ has the property that for all $a,b,g \in G$, $\begin{align} > a \sim b &\implies \textcolor{Thistle}{({\dagger}) \ ga \sim gb} \text{ and } \textcolor{Skyblue}{({\dagger} {\dagger}) \ ag \sim bg}.\\ > \end{align}$ > 3. The elements of $G/{\sim}$ are exactly the [[coset|(left, =right) cosets]] of a [[normal subgroup]] $N=[e_{G}]_{\sim}$. > > Indeed, there is a [[bijection]] $\left\{ \begin{align} > &\text{equivalence relations } \sim \\ > & \text{that satsify } \textcolor{Thistle}{({\dagger})} \text{ and } \textcolor{Skyblue}{({\dagger}{\dagger})} > \end{align} \right\} \leftrightarrow \{ \text{normal subgroups of} \ G \}.$ > Furthermore, if (1) holds, then the [[group]] structure on $G / {\sim}$ is uniquely determined with [[binary operation]] $[a] \bullet [b]:=[ab]$, or equivalently $aN \bullet bN := (ab)N$. The following [[universal property]] follows: >> [!theorem] Theorem. ([[characterization of quotienting a group|universal property of quotient groups]]) >> Let $N$ be a [[normal subgroup]] of a [[group]] $G$, inducing (via the theorem statement) an [[equivalence relation]] $\sim$. >> > >Let $\varphi: G \to G'$ be a [[group homomorphism]] satisfying one of the following, manifestly equivalent conditions: > >- $g \sim h \implies \varphi(g)=\varphi(h)$; > >- $gN=hN \implies \varphi(g)=\varphi(h)$; > > - $N \subset \text{ker }\varphi$. [^1] > > > >Then there exists a unique [[group homomorphism|homomorphism]] $\overline{\varphi}:G / N \to H$ such that the following diagram commutes: >> > >```tikz > >\usepackage{tikz-cd} >> > >\begin{document} >> > >\begin{tikzcd} >> G \arrow{d}[swap]{\pi} \arrow{r}{\varphi} & G' \\ > >G/N \arrow{ur}[swap]{\exists !\, \bar{\varphi}} > >\end{tikzcd} >> > >\end{document} > >``` >> > >More precisely, $G / N=G /{\sim}$ is [[coslice category|coslice]]-[[terminal object|initial]] in $\mathsf{Grp}^{G}$ with respect to the [[subcategory]] obtained by keeping those [[group homomorphism|homomorphisms]] $\varphi$ satisfying $g \sim h \implies \varphi(g) = \varphi(h)$ (equivalently $gN=hN \implies \varphi(g)=\varphi(h)$). > > Conversely, a map $\overline{\varphi}:G / N \to G'$ out of the quotient $G / N$ induces a unique $\varphi:G \to G'$ causing the diagram to commute. ^90a179 > [!proof]- Proof. ([[characterization of quotienting a group]]) > > **$(1) \iff (2)$.** Recall the notion of [[quotient set]] and the [[universal property]] [[universal property of quotient sets|it satisfies]]. How can we investigate this situation in the [[category]] $\mathsf{Grp}$? We look for an [[equivalence relation]] $\sim$ on the (set underlying) a [[group]] $G$, and seek a [[binary operation]] on $G /{\sim}$ that would turn it into [[group]]. > > What should multiplication look like in $G /{\sim}$? Well, in order for $\pi$ to be a [[group homomorphism|homomorphism]] we must have, at least formally, $[g] \cdot [h]=\pi(g)\pi(h)=\pi(gh)=[gh].$But is this [[well-defined]]? That would depend on the nature of $\sim$... > - For the operation to be [[well-defined]] 'in the first factor', we need that if $[g]=[g']$ then $[gh]=[g'h]$ regardless of $h$. That is, $\text{For all } h \in G, g \sim g' \implies gh \sim g'h.$ > - For the operation to be [[well-defined]] 'in the second factor', we need that if $[g]=[g']$ then $[hg]=[hg']$. That is, $\text{For all } h \in G, g \sim g' \implies hg \sim hg'.$We have shown that $\sim$ necessarily satisfies these two properties. But satisfying them is sufficient too: if both conditions are satisfied then the operation $[a] \bullet [b]:=[ab]$ > is [[well-defined]] and yields a [[group]] with identity $[e_{G}]$, inverses $[g ^{-1}]$, and associativity inherited from $G$. Thus $(1) \iff (2)$. > > > **$(2) \iff (3)$ and Bijection.** This follows from the [[characterization of cosets]]. > > > > > **Universal Property.** To see that the [[universal property]] is satisfied, let $N$ be a [[normal subgroup]] of $G$ with corresponding [[equivalence relation]] $\sim$. Let $\varphi$ be such that $a \sim a'$ implies $\varphi(a)=\varphi(a')$. Then since the *set* $G /{\sim}$ satisfies the corresponding [[universal property]] in $\mathsf{Set}$, we know there exists a unique injective set-function $\tilde{\varphi}: G /{\sim} \to G'$ > ([[universal property of quotient sets|well]])-defined as $\tilde{\varphi}([g])=\varphi(g)$. We just need to show $\tilde{\varphi}$ is a [[group homomorphism]], which it clearly is: $\tilde{\varphi}([g] \bullet [h])=\tilde{\varphi}([gh])=\varphi(gh)=\varphi(g)\varphi(h)=\tilde{\varphi}([g]) \tilde{\varphi}([h]).$ ----- #### [^1]: Indeed, $N$ belongs to the [[kernel of a group homomorphism|kernel]] of $\varphi$ if and only if $\varphi(n)=e_{G'}$ for all $n \in N$, which in turn happens if and only if for all $a,b \in G$, $ab ^{-1} \in N \implies \varphi(ab ^{-1})=e_{G'}$, i.e., $ab ^{-1} \in N \implies \varphi(a)=\varphi(b)$, i.e., $a \sim b \implies \varphi(a) = \varphi(b)$. ---- #### References > [!backlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as Tag > [!frontlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM outgoing([[]]) FLATTEN file.tags GROUP BY file.tags as Tag