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  "body": "\n## Statement\n\nLet $G$ be an [[abelian group|082GHNPJ53]] and let $H$ be a [[subgroup|0X1JXT47GA]] of $G$.\nThen $H$ is [[normal|0CB0DX5BVV]] in $G$.\n\n## Remarks\n\nFor $g \\in G$ and $h \\in H$, commutativity gives $ghg^{-1} = hgg^{-1} = h \\in H$.\nThe converse is false: in the quaternion group $Q_8$ each subgroup is normal, but $Q_8$ is not abelian.\nMore generally, each subgroup of the center of any group is normal.\n\n## Explanations\n\n### L1 human\n\nIn an abelian group, conjugation does nothing, because $ghg^{-1} = h$.\nSo a subgroup can never be moved by conjugation.\nThus all subgroups are normal.\n",
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