[[Abelian group]]
# Abelianization

The **abelianization** $G^\mathrm{ab}$ of a group $G$ is the largest abelian [[Quotient group|quotient]] of $G$. #m/def/group
For a group $G$ is abelianized by taking the quotient with the [[Commutator subgroup]] $[G,G]$.
$$
\begin{align*}
G^\mathrm{ab} = G / [G,G]
\end{align*}
$$

## Main theorem

Let $N \trianglelefteq G$. The [[quotient group]] $G / N$ is [[Abelian group|abelian]] iff $[G,G] \trianglelefteq N$. #m/thm/group

> [!check]- Proof
> $G / N$ is abelian iff $[a,b]N = aba^{-1}b^{-1}N = eN = N$ for all $a,b \in G$,
> and the latter holds iff $[a,b] \in N$ for all $a,b \in G$.
> <span class="QED"/>


## Universal property

Abelianization has a unique extension to a [[functor]] $(-)^\mathrm{ab} : \Grp \to \Ab$ from [[Category of groups]] into [[Category of abelian groups]] so that the projection becomes a [[natural transformation]] $\pi : 1 \Rightarrow (-)^\mathrm{ab} : \Grp \to \Grp$.
This is done by defining $(G^\mathrm{{ab}}, \pi_{G})$ with the help of the following universal property:

$G^\mathrm{ab}$ is abelian.
If $H$ is an abelian group and $\varphi \in \Grp(G,H)$ is a homomorphism,
then there exists a unique $\bar \varphi \in \Ab (G,H)$ such that $\varphi = \bar \varphi \pi_{G}$, 
i.e. the following diagram commutes
<p align="center"><img align="center" src="
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#invert" alt="https://q.uiver.app/#q=WzAsNSxbMiwyLCJcXG1hdGhybSBJIEgiXSxbMiwwLCJcXG1hdGhybSBJIEdeXFxtYXRocm17YWJ9Il0sWzAsMCwiRyJdLFs0LDAsIkdeXFxtYXRocm17YWJ9Il0sWzQsMiwiSCJdLFsyLDEsIlxccGlfRyJdLFsxLDAsIlxcbWF0aHJtIEkgXFxiYXJcXHZhcnBoaSJdLFsyLDAsIlxcdmFycGhpIiwyXSxbMyw0LCJcXGJhciBcXHZhcnBoaSJdXQ==&amp;macro_url=%5CDeclareMathOperator%7B%5Cid%7D%7Bid%7D" /></p>

> [!check]- Proof
> $G^\mathrm{ab}$ is abelian by construction.
> By properties of the [[Kernel of a homomorphism into an abelian group]],
> the universal property is equivalent to [[Quotient group#Universal property|that of the quotient group]].
> <span class="QED"/>

This of course forms a [[Free-forgetful adjunction]]

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