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  "body": "\n## Statement\n\nLet $n \\geq 5$.\nThen the [[symmetric group|0515A6D2Y1]] $S_n$ is not [[solvable|0FXJ7YFMMX]].\n\n## Remarks\n\nThe group $A_n$ is [[simple|0ARPRJQWZ2]] and not abelian, so its commutator subgroup is $A_n$, and the derived series of $A_n$ is constant.\nSo $A_n$ is not solvable, and then $S_n$ is not solvable, because [[subgroups of solvable groups are solvable|05Y0F9M0TS]].\nThe groups $S_1, S_2, S_3$ and $S_4$ are solvable.\nBy Galois theory, this is the reason why the general polynomial equation of degree $n \\geq 5$ is not solvable by radicals.\n\n## Explanations\n\n### L1 human\n\nFrom five letters on, the even permutations form a simple group that is not abelian.\nSuch a group cannot be cut into abelian layers.\nSo the full symmetric group cannot be cut into abelian layers either.\n",
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