= Abelianization
{title2=$G^{\mathrm{ab}}$}
{wiki}
= Abelianisation
{synonym}
The abelianization of a <group> $G$ is the <quotient group>
$$
G^{\mathrm{ab}}=G/G',\qquad q:G\longrightarrow G^{\mathrm{ab}},\quad q(g)=gG',
$$
where $G'$ is the <commutator subgroup>. Its elements are <cosets>, and its operation is $(gG')(hG')=ghG'$. The result is an <abelian group>: the two products $gh$ and $hg$ differ by a <group commutator>, which disappears in the <quotient group>.
For the <symmetric group> $S_3$, the <sign of a permutation> realizes $S_3^{\mathrm{ab}}\cong C_2$, as shown in <commutator subgroup of the symmetric group on three letters>. For the <dihedral group> of a <square>, <abelianization of the square symmetry group> gives $C_2\times C_2$: two independent yes-or-no pieces of information survive. An <abelian group> already has $G'=\{e\}$, so its <abelianization> is isomorphic to itself. At the other extreme, a <perfect group> has trivial <abelianization>.
Calling this the largest <abelian quotient> means that every <group homomorphism> into any <abelian group> factors uniquely through $q$. It does not require $G$ to be finite, or compare the sizes of unrelated <abelian groups>. The <universal property of abelianization> makes this precise. It lets us solve all such mapping problems in one simpler <group>: any invariant that respects multiplication and takes values in an <abelian group> sees only $G^{\mathrm{ab}}$.
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