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The abelianization of a group is the quotient $G^{\mathrm{ab}}=G/G^{\prime }$ by its commutator subgroup. It is the largest abelian quotient of $G$.

For odd $n\ge 3$, the dihedral group $D_{2n}$ of order $2n$ has rotation generator $r$ and reflection generator $s$, with $srs=r^{-1}$. The group commutator $[r,s]=r^{-2}$ generates the rotation subgroup, while the quotient by that subgroup is $C_2$. Hence the commutator subgroup is exactly $⟨r⟩$ and the abelianization is $C_2$. The only abelian quotients are $C_2$ and the trivial group.

Suppose $1\to X\to \mathcal{G}\to \Gamma \to 1$ is an exact sequence of pro-p groups, with $X$ abelian and $\Gamma \cong {\mathbb{Z}}_p$. Conjugation gives $X$ a compact Galois module structure. The closed commutator subgroup is $(\gamma -1)X$: its image is closed by compactness, and after quotienting by it, a lift of $\gamma$ centralizes $X$ and topologically generates the remaining quotient. Hence $0\to X_{\Gamma }\to {\mathcal{G}}^{\mathrm{ab}}\to \Gamma \to 0$ is exact.

An abelian quotient is a quotient group $G/N$ that is an abelian group. It exists only for a normal subgroup $N$, and it is abelian exactly when $N$ contains the commutator subgroup. Thus every abelian quotient factors through the abelianization $G/[G,G]$.

A finite nonabelian simple group is a finite group that is a nonabelian group and a simple group. Every such group is a perfect group because its commutator subgroup is a nontrivial normal subgroup.

The group commutator measures the failure of two elements to commute. Under the convention $[x,y]=x^{-1}{y}^{-1}xy$, it is the identity exactly when $xy=yx$.

Let $\pi :G\to G/[G,G]$ be the abelianization map and let $A$ be a finite $K$-approximate group. Suppose a coset progression $HP(x_1,\dots ,x_r;L_1,\dots ,L_r)$ lies in $\pi (A^4)$ and has size at least $\delta |\pi (A)|$. ThenA section of $\pi$ is multiplicative up to a bounded power of $A$ inside the commutator subgroup; successively separating the subgroup and progression coordinates proves the inclusion needed for this estimate.

A group is metabelian when its commutator subgroup is abelian, equivalently when its derived length is at most two.

Every one-dimensional character kills the commutator subgroup and therefore factors through the abelianization. Conversely, every character of the abelianization pulls back to a one-dimensional character of the group.

The image of a one-dimensional representation lies in the abelian group ${\mathbb{C}}^{\times }$, so its kernel contains the commutator subgroup of the represented group.