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  "body": "\n## Definition\n\nLet $G$ and $H$ be [[groups|G5S1M9]], let $e_H$ be the identity element of $H$, and let $\\varphi \\colon G \\to H$ be a [[group homomorphism|0JYTG5K9Y4]].\nThe kernel of $\\varphi$ is the set $\\ker \\varphi = \\{g \\in G : \\varphi(g) = e_H\\}$.\n\n## Remarks\n\nThe kernel is a [[normal subgroup|0CB0DX5BVV]] of $G$.\nThe kernel of the map $\\mathbb{Z} \\to \\mathbb{Z}/n\\mathbb{Z}$, $k \\mapsto k + n\\mathbb{Z}$, is $n\\mathbb{Z}$.\nThe kernel of the determinant map from $\\mathrm{GL}_n(\\mathbb{R})$ to $\\mathbb{R}^\\times$ is the special linear group $\\mathrm{SL}_n(\\mathbb{R})$.\n\n## Explanations\n\n### L1 human\n\nThe kernel is the part of the group that the homomorphism forgets: everything in it goes to the identity.\nIf the kernel is only the identity, the homomorphism forgets nothing, and it is injective.\nFor the map that sends an integer to its remainder modulo $3$, the kernel is the set of multiples of $3$.\n",
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