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  "body": "\n## Statement\n\nLet $G$ and $H$ be [[groups|G5S1M9]], let $e_G$ be the identity element of $G$, and let $\\varphi \\colon G \\to H$ be a [[group homomorphism|0JYTG5K9Y4]].\nThen $\\varphi$ is injective if and only if $\\ker \\varphi = \\{e_G\\}$.\n\n## Remarks\n\nIf $\\varphi(a) = \\varphi(b)$, then $\\varphi(a^{-1}b) = e_H$, so $a^{-1}b \\in \\ker \\varphi$.\nMore generally, $\\varphi(a) = \\varphi(b)$ exactly when $a$ and $b$ lie in the same coset of $\\ker \\varphi$.\n\n## Explanations\n\n### L1 human\n\nTo check that a homomorphism is injective, you only need to check one fibre: the elements that go to the identity.\nIf only the identity goes to the identity, then no two elements can go to the same place.\nThis is much easier than comparing all pairs of elements.\n",
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