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  "body": "\n## Statement\n\nLet $G$ and $H$ be [[groups|G5S1M9]] and let $\\varphi \\colon G \\to H$ be a [[group homomorphism|0JYTG5K9Y4]].\nThen the [[kernel|0W88CEGQP1]] $\\ker \\varphi$ is a [[normal subgroup|0CB0DX5BVV]] of $G$.\n\n## Remarks\n\nFor $g \\in G$ and $k \\in \\ker \\varphi$, $\\varphi(gkg^{-1}) = \\varphi(g) \\, e_H \\, \\varphi(g)^{-1} = e_H$.\nConversely, each normal subgroup $N$ of $G$ is the kernel of the projection $G \\to G/N$.\nThus the normal subgroups of $G$ are exactly the kernels of the homomorphisms from $G$.\n\n## Explanations\n\n### L1 human\n\nIf an element goes to the identity, then each of its conjugates also goes to the identity.\nSo the kernel is closed under conjugation, which is the definition of a normal subgroup.\nThis is one reason why normal subgroups are important.\n",
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