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  "body": "\n## Statement\n\nLet $G$ be a [[group|G5S1M9]] and let $H$ be a [[subgroup|0X1JXT47GA]] of $G$ of [[index|0DTXBQPA2S]] $2$.\nThen $H$ is [[normal|0CB0DX5BVV]] in $G$.\n\n## Remarks\n\nFor $g \\notin H$, the left cosets are $H$ and $gH = G \\setminus H$, and the right cosets are $H$ and $Hg = G \\setminus H$.\nThus $gH = Hg$ for each $g \\in G$, and [[the coset criterion|01E7SWYJX3]] applies.\nFor example, the alternating group $A_n$ has index $2$ in the symmetric group $S_n$, and thus it is normal.\n\n## Explanations\n\n### L1 human\n\nWhen a subgroup has only two cosets, one coset is the subgroup and the other is everything else.\nThis is true for the left cosets and for the right cosets.\nSo the left and right cosets agree, and the subgroup is normal.\n",
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