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  "body": "\n## Statement\n\nLet $n \\geq 2$.\nThe [[alternating group|0Z5905Q73D]] $A_n$ has [[order|0B85S2V2DR]] $n!/2$, and $[S_n : A_n] = 2$.\n\n## Remarks\n\nThe sign map is a surjective homomorphism onto a group of order $2$ with kernel $A_n$, so $S_n / A_n$ has order $2$.\nAnother proof: the map $\\sigma \\mapsto (1\\ 2) \\sigma$ is a bijection from the even permutations to the odd permutations.\nFor example, $|A_4| = 12$ and $|A_5| = 60$.\n\n## Explanations\n\n### L1 human\n\nHalf of all permutations are even.\nTo see this, multiply by one fixed swap: this changes each even permutation into an odd one, and back.\nSo $S_n$ splits into two halves of the same size.\n",
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