= Universal property of a quotient group
{title2=$\varphi=\bar\varphi\circ\pi$}
Let $N$ be a <normal subgroup> of $G$, and let $\pi:G\to G/N$ be the <quotient map of groups>. If a <group homomorphism> $\varphi:G\to H$ has <group kernel> containing $N$, there is exactly one <group homomorphism> $\bar\varphi:G/N\to H$ with $\varphi=\bar\varphi\circ\pi$. Its value is forced: $\bar\varphi(gN)=\varphi(g)$. Every measurement by a <group homomorphism> that ignores $N$ can thus be performed on the <quotient group> directly. This is the <universal property> of the construction.
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