= Commutator subgroup
{title2=$G'=[G,G]$}
{wiki}
= Derived subgroup
{synonym}
The commutator subgroup of a <group> $G$ is the <subgroup> generated by all its <group commutators>:
$$
G'=[G,G]=\langle x^{-1}y^{-1}xy:x,y\in G\rangle.
$$
Here and throughout, $[x,y]=x^{-1}y^{-1}xy$. A <group commutator> is the <identity element> precisely when $x$ and $y$ commute. The generated <subgroup> includes finite products of <group commutators> and their <inverse elements>; an element of $G'$ need not itself be a single <group commutator>. Thus $G'=\{e\}$ precisely when $G$ is an <abelian group>.
The useful idea is to record exactly what must disappear to make multiplication commutative. The identity $xy=yx[x,y]$ shows how a <group commutator> corrects a change in multiplication order. Setting every such correction equal to $e$ produces the <abelianization> $G/G'$, while preserving every piece of information that a <group homomorphism> to an <abelian group> can detect. This connects a local comparison of two operations to a canonical <quotient group> of the whole <group>.
The commutator subgroup is a <characteristic subgroup>, hence a <normal subgroup>: a <group automorphism> $\alpha$ sends $[x,y]$ to $[\alpha(x),\alpha(y)]$. For any <normal subgroup> $N$,
$$
G/N\text{ is abelian}\quad\Longleftrightarrow\quad G'\subseteq N.
$$
Indeed, the <group commutator> of $xN$ and $yN$ is $[x,y]N$. This makes $G'$ the smallest <normal subgroup> whose <quotient group> is an <abelian group>. See https://kconrad.math.uconn.edu/blurbs/grouptheory/subgpseries1.pdf[Conrad, Subgroup series I, section 4: commutators and abelian quotients].
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