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  "body": "\n## Example\n\nLet $n \\ge 2$. The [[sign|0SK90238BH]] is a homomorphism $\\operatorname{sgn} : S_n \\to \\{1, -1\\}$, where $\\{1, -1\\}$ is a group under multiplication.\nA permutation has sign $1$ when it is a product of an even number of transpositions.\nThus the kernel of $\\operatorname{sgn}$ is the set of even permutations. This is the [[alternating group|0Z5905Q73D]] $A_n$.\nThus $A_n$ is a [[normal subgroup|0C9ZKYSPYR]] of $S_n$.\n\n## Remarks\n\nThe sign is onto, so $A_n$ has index $2$ in $S_n$.\n",
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