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  "body": "\n## Statement\n\nLet $G$ be a [[group|G5S1M9]] and let $H$ be a [[characteristic subgroup|0KABPMRHM2]] of $G$.\nThen $H$ is a [[normal subgroup|0CB0DX5BVV]] of $G$.\n\n## Remarks\n\nEach [[inner automorphism|0RN00NTM3S]] $c_g$ is an automorphism, so $gHg^{-1} = c_g(H) = H$ for each $g \\in G$.\nThe converse is false: in the Klein four-group $V = \\{e, a, b, ab\\}$, each subgroup is normal, but an automorphism can swap $\\{e, a\\}$ and $\\{e, b\\}$.\n\n## Explanations\n\n### L1 human\n\nNormal means fixed by the inner automorphisms, and characteristic means fixed by all automorphisms.\nAll automorphisms include the inner ones.\nSo a characteristic subgroup is normal.\n",
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