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  "body": "\n## Proof\n\n1. Let $\\varphi : G \\to H$ be a group homomorphism with [[kernel|0Z424HK8CQ]] $K$.\n2. Suppose that $\\varphi$ is injective, and let $k \\in K$. [[$\\varphi(e) = e$|08FKQD7NJ3]], so $\\varphi(k) = \\varphi(e)$. Thus $k = e$, and $K = \\{e\\}$.\n3. Now suppose that $K = \\{e\\}$, and let $a, b \\in G$ with $\\varphi(a) = \\varphi(b)$.\n4. Then $\\varphi(a^{-1}b) = \\varphi(a)^{-1}\\varphi(b) = e$, so $a^{-1}b \\in K$.\n5. Thus $a^{-1}b = e$, so $a = b$, and $\\varphi$ is injective.\n\n## Idea\n\nTwo elements have the same image exactly when their quotient is in the kernel.\n",
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