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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]] and let $H$ be a [[subgroup|0X1JXT47GA]] of $G$.\nThe subgroup $H$ is characteristic in $G$ when $\\alpha(H) = H$ for each [[automorphism|0K1SCPBNKJ]] $\\alpha$ of $G$.\n\n## Remarks\n\nIt is equivalent to ask $\\alpha(H) \\subseteq H$ for each automorphism $\\alpha$, because then also $\\alpha^{-1}(H) \\subseteq H$.\nThe [[centre|0DZ93493VN]] and the [[commutator subgroup|0ASGSTXSBW]] of a group are characteristic.\nA characteristic subgroup of a normal subgroup of $G$ is normal in $G$.\n\n## Explanations\n\n### L1 human\n\nA characteristic subgroup is a subgroup that no symmetry of the group can move.\nUsually such a subgroup has a description that uses only the group operation, like \"the elements that commute with everything\".\nAny renaming of the elements that keeps the products correct must keep such a subgroup in place.\n",
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