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text: 'A *normal subgroup* $N$ of group $G$ is one which is closed under [4gk conjugation]: for all $h \\in G$, it is the case that $\\{ h n h^{-1} : n \\in N \\} = N$.\nIn shorter form, $hNh^{-1} = N$.\n\nSince [4gk conjugacy is equivalent to "changing the worldview"], a normal subgroup is one which "looks the same from the point of view of every element of $G$".\n\nA subgroup of $G$ is normal if and only if it is the [49y kernel] of some [-47t] from $G$ to some group $H$. ([4h7 Proof.])\n\n%%%knows-requisite([category_theory_equaliser]):\nFrom a category-theoretic point of view, the kernel of $f$ is an equaliser of an arrow $f$ with the zero arrow; it is therefore universal such that composition with $f$ yields zero.\n%%%\n\n[todo: why are they interesting]',
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