= Universal property of abelianization
{title2=$f=\bar f\circ q$}
For every <abelian group> $A$ and <group homomorphism> $f:G\to A$, there is a unique <group homomorphism> $\bar f:G^{\mathrm{ab}}\to A$ such that
$$
f=\bar f\circ q,\qquad \bar f(gG')=f(g).
$$
This is a <universal property>. To see why it holds, compute $f([x,y])=[f(x),f(y)]=e_A$, using <commutativity> in $A$. Hence $G'\subseteq\ker f$, and $f$ is constant on each <coset> of $G'$. The displayed formula is therefore well-defined and preserves multiplication. It is unique because $q$ is <surjective>. This is an instance of factoring a <group homomorphism> through a <quotient group>; see https://kconrad.math.uconn.edu/blurbs/linmultialg/universalmapping.pdf[Conrad, Universal mapping properties, theorem 3.4].
In particular, every <abelian quotient> $G/N$ comes from a <surjective group homomorphism> $G^{\mathrm{ab}}\to G/N$, $gG'\mapsto gN$. For $S_3$, any <group homomorphism> to an <abelian group> is determined by the common image $a$ of its three <transpositions>, with $a^2=e_A$. Every <even permutation> must map to $e_A$. Thus <sign of a permutation> captures all possible information of this kind, including maps with infinite <abelian groups> as their targets.
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