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  "body": "\n## Statement\n\nLet $n \\geq 2$.\nThe map $\\operatorname{sgn} \\colon S_n \\to \\{1, -1\\}$ is a surjective [[group homomorphism|0JYTG5K9Y4]], where $\\{1, -1\\}$ is a group under multiplication.\nThat is, $\\operatorname{sgn}(\\sigma \\tau) = \\operatorname{sgn}(\\sigma) \\operatorname{sgn}(\\tau)$ for all $\\sigma, \\tau \\in S_n$.\n\n## Remarks\n\nIf $\\sigma$ is a product of $m$ transpositions and $\\tau$ is a product of $l$ transpositions, then $\\sigma \\tau$ is a product of $m + l$ transpositions.\nThe map is surjective because $\\operatorname{sgn}((1\\ 2)) = -1$.\nIts [[kernel|0W88CEGQP1]] is the [[alternating group|0Z5905Q73D]] $A_n$.\n\n## Explanations\n\n### L1 human\n\nWhen you do two permutations one after the other, you join their chains of swaps.\nThe number of swaps adds, so the signs multiply.\nThis is exactly the rule of a homomorphism.\n",
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