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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 map $g \\ker \\varphi \\mapsto \\varphi(g)$ is a well-defined [[isomorphism|08MG7H8JW5]] from the [[quotient group|0KYX2N69J0]] $G / \\ker \\varphi$ to the [[image|0S21GNW9ZB]] $\\operatorname{im} \\varphi$.\n\n## Remarks\n\nHere $\\ker \\varphi$ is normal by [[this fact|0C9ZKYSPYR]], so the quotient exists.\nIf $\\varphi$ is surjective, then $H \\cong G / \\ker \\varphi$.\nFor example, the determinant gives $\\mathrm{GL}_n(\\mathbb{R}) / \\mathrm{SL}_n(\\mathbb{R}) \\cong \\mathbb{R}^\\times$.\nFor finite $G$, a consequence is $|G| = |\\ker \\varphi| \\cdot |\\operatorname{im} \\varphi|$.\n\n## Explanations\n\n### L1 human\n\nA homomorphism collapses each coset of its kernel to one point of the image.\nAfter this collapse, nothing else is lost: the quotient by the kernel and the image are the same group.\nFor example, reduction modulo $n$ shows that $\\mathbb{Z}/n\\mathbb{Z}$ is the image of $\\mathbb{Z}$.\n",
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