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  "body": "\n## Statement\n\nLet $G$ and $H$ be [[groups|G5S1M9]], let $N$ be a [[normal subgroup|0CB0DX5BVV]] of $G$, and let $\\pi \\colon G \\to G/N$ be the projection $\\pi(g) = gN$ to the [[quotient group|0KYX2N69J0]].\nLet $\\varphi \\colon G \\to H$ be a [[group homomorphism|0JYTG5K9Y4]] with $N \\subseteq \\ker \\varphi$.\nThen there is exactly one group homomorphism $\\bar{\\varphi} \\colon G/N \\to H$ with $\\bar{\\varphi} \\circ \\pi = \\varphi$, and it is given by $\\bar{\\varphi}(gN) = \\varphi(g)$.\n\n## Remarks\n\nThe map $\\bar{\\varphi}$ is well defined because $\\varphi$ is constant on each coset of $N$.\nThe image of $\\bar{\\varphi}$ is the image of $\\varphi$, and the kernel of $\\bar{\\varphi}$ is $(\\ker \\varphi)/N$.\nWith $N = \\ker \\varphi$, this gives the [[first isomorphism theorem|0CXXJNG5B6]].\n\n## Explanations\n\n### L1 human\n\nA homomorphism that sends all of $N$ to the identity cannot tell apart elements of the same coset of $N$.\nSo it already makes sense on the cosets, that is, on the quotient group.\nThis is the standard way to define a homomorphism out of a quotient group.\n",
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  "name": "Universal property of the quotient group",
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