[[Group theory MOC]]
# Commutator subgroup

The **commutator subgroup** $[G,G]$ is a [[normal subgroup]] of $G$ generated by the [[group commutator]] of all elements, pairwise. #m/def/group 
$$
\begin{align*}
[G,G] = \langle [a,b] = aba^{-1}b^{-1} : a,b \in G \rangle \trianglelefteq G
\end{align*}
$$

> [!check]- Proof of normal subgroup
> $[G,G]$ is a subgroup by construction.
> Let $g \in [G,G]$.
> Then for any conjugate $y = xgx^{-1}$ it follows $yg^{-1} = xgx^{-1}g^{-1} = [x,g]$,
> so $yg^{-1} \in [G,G]$ and thus $yg^{-1}g = y \in [G,G]$.
> Therefore $[G,G]$ is a normal subgroup.
> <span class="QED"/>

Wikipedia notes

> \[the commutator subgroup\] is stable under every endomorphism of $G$: that is, $[G,G]$ is a [fully characteristic subgroup](https://en.wikipedia.org/wiki/Fully_characteristic_subgroup "Fully characteristic subgroup") of $G$, a property considerably stronger than normality.

## Properties

- A quotient with the commutator subgroup of $G$ is called an [[Abelianization]] of $G$.

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