Let $[G,G] \trianglelefteq G$ denote the [normal subgroup](/page/Normal%20Subgroup) of $G$ generated by all commutators $[g,h]$ with $g,h \in G$. Define the abelianisation of $G$ to be the [quotient group](/theorems/790)
and let $\pi: G \to G^{\mathrm{ab}}$ be the quotient homomorphism
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\begin{align*}
\pi(g) := g [G,G].
\end{align*}
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If $A$ is an [abelian group](/page/Abelian%20Group) and $\varphi: G \to A$ is a [group homomorphism](/page/Group%20Homomorphism), then there exists a unique homomorphism of abelian groups $\overline{\varphi}: G^{\mathrm{ab}} \to A$ such that