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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]].\nThe derived series of $G$ is the sequence of subgroups $G^{(0)} = G$ and $G^{(i+1)} = [G^{(i)}, G^{(i)}]$ for $i \\geq 0$.\nHere $[H, H]$ is the [[commutator subgroup|0ASGSTXSBW]] of $H$.\n\n## Remarks\n\nEach $G^{(i)}$ is a [[characteristic subgroup|0KABPMRHM2]] of $G$, and $G^{(i)} / G^{(i+1)}$ is abelian.\nThe group $G$ is [[solvable|0FXJ7YFMMX]] if and only if $G^{(n)} = \\{e\\}$ for some $n$, and the least such $n$ is the derived length.\nFor $S_4$, the derived series is $S_4 \\supseteq A_4 \\supseteq V \\supseteq \\{e\\}$.\nFor $n \\geq 5$, the derived series of $S_n$ is $S_n \\supseteq A_n \\supseteq A_n \\supseteq \\cdots$, so it never reaches $\\{e\\}$.\n\n## Explanations\n\n### L1 human\n\nEach step of the derived series takes the commutator subgroup of the last step.\nSo each step removes an abelian layer from the group.\nIf you reach the identity, the group is solvable.\n",
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