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  "body": "\n## Proof\n\nWe show that the formula $\\bar{\\varphi}(gN) = \\varphi(g)$ is well defined, that it gives a homomorphism, and that no other homomorphism works.\n\n1. Let $N$ be a [[normal subgroup|0W77CVBCHQ]] of $G$, and let $\\varphi \\colon G \\to H$ be a [[group homomorphism|0HV9Y7WG5R]] with $N \\subseteq \\ker \\varphi$. Let $e_G$ and $e_H$ be the identities.\n2. The [[quotient group|0YYWCNTPWR]] $G/N$ is the set of [[left cosets|05FD317V2W]] $gN$, with $(aN)(bN) = (ab)N$. The projection is $\\pi(g) = gN$.\n3. Let $g, g' \\in G$ with $gN = g'N$. Since $e_G \\in N$, $g' = g' e_G \\in g'N = gN$. So $g' = gn$ for some $n \\in N$.\n4. By step 1, $n \\in \\ker \\varphi$, so $\\varphi(n) = e_H$ by the definition of the [[kernel|0Z424HK8CQ]]. Hence $\\varphi(g') = \\varphi(g) \\varphi(n) = \\varphi(g)$.\n5. By steps 3 and 4, the value $\\varphi(g)$ depends only on the coset $gN$. So $\\bar{\\varphi}(gN) = \\varphi(g)$ is a well defined map $\\bar{\\varphi} \\colon G/N \\to H$.\n6. For $a, b \\in G$, step 2 gives $\\bar{\\varphi}((aN)(bN)) = \\bar{\\varphi}((ab)N) = \\varphi(ab) = \\varphi(a) \\varphi(b) = \\bar{\\varphi}(aN) \\bar{\\varphi}(bN)$. So $\\bar{\\varphi}$ is a group homomorphism.\n7. For $g \\in G$, $\\bar{\\varphi}(\\pi(g)) = \\bar{\\varphi}(gN) = \\varphi(g)$. So $\\bar{\\varphi} \\circ \\pi = \\varphi$, and a homomorphism with this property exists.\n8. Let $\\psi \\colon G/N \\to H$ be any group homomorphism with $\\psi \\circ \\pi = \\varphi$. Each element of $G/N$ has the form $gN = \\pi(g)$ for some $g \\in G$.\n9. For such an element, $\\psi(gN) = \\psi(\\pi(g)) = \\varphi(g) = \\bar{\\varphi}(gN)$. So $\\psi = \\bar{\\varphi}$.\n10. By steps 5 to 9, there is exactly one homomorphism $\\bar{\\varphi}$ with $\\bar{\\varphi} \\circ \\pi = \\varphi$, and it is given by $\\bar{\\varphi}(gN) = \\varphi(g)$.\n\n## Idea\n\nTwo elements in the same coset differ by an element of $N$, and $\\varphi$ sends that element to the identity. So $\\varphi$ takes one value on each coset, which defines $\\bar{\\varphi}$. Uniqueness holds because $\\pi$ is surjective: every coset is $\\pi(g)$, so the value on it is forced to be $\\varphi(g)$.\n",
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