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  "body": "\n## Definition\n\nLet $n$ be a positive integer.\nThe alternating group $A_n$ is the set of the [[even permutations|0SQTMQTSRZ]] in $S_n$, with composition as the operation.\n\n## Remarks\n\nFor $n \\geq 2$, $A_n$ is the [[kernel|0W88CEGQP1]] of the [[sign homomorphism|0JEJJCB0EX]], so it is a [[normal subgroup|0CB0DX5BVV]] of $S_n$.\nFor $n \\geq 2$, $A_n$ is also the [[commutator subgroup|0ASGSTXSBW]] of $S_n$.\nThe group $A_3$ is cyclic of order $3$, and $A_4$ has order $12$.\nThe group $A_n$ is abelian if and only if $n \\leq 3$.\n\n## Explanations\n\n### L1 human\n\nThe alternating group keeps only the even permutations.\nThe product of two even permutations is even, so these form a group.\nFor $n \\geq 5$ it is one of the most important examples of a simple group.\n",
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