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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\n### L2 agent\n\nA subgroup $H$ is normal when $gH = Hg$ for each $g \\in G$. In general this is a real condition. For a subgroup of index $2$ it is automatic.\n\nThe reason is that two cosets leave no choice:\n\n1. The left cosets are $H$ and one other coset. Since the cosets partition $G$, the other coset is $G \\setminus H$.\n2. In the same way, the right cosets are $H$ and $G \\setminus H$.\n3. If $g \\in H$, then $gH = H = Hg$.\n4. If $g \\notin H$, then $gH = G \\setminus H = Hg$.\n\nA worked example with $G = S_3$ and $H = A_3 = \\{e, (1\\,2\\,3), (1\\,3\\,2)\\}$:\n\n1. $|S_3| = 6$ and $|A_3| = 3$, so the index is $2$.\n2. For $g = (1\\,2)$, the left coset $gA_3$ is the set of the three transpositions.\n3. The right coset $A_3g$ is the same set. Thus $gA_3 = A_3g$.\n\nThe index two condition is sharp. The subgroup $\\{e, (1\\,2)\\}$ of $S_3$ has index $3$, and it is not normal: $(1\\,2\\,3)H = \\{(1\\,2\\,3), (1\\,3)\\}$, but $H(1\\,2\\,3) = \\{(1\\,2\\,3), (2\\,3)\\}$.\n\nStandard examples are $A_n$ in $S_n$, the rotations in a dihedral group, and $2\\mathbb{Z}$ in $\\mathbb{Z}$. A related fact is that a subgroup whose index is the smallest prime divisor of $|G|$ is also normal. The proof of that fact needs a group action.\n",
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