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  "body": "\n## Statement\n\nLet $G$ be a [[solvable group|0FXJ7YFMMX]] and let $H$ be a [[subgroup|0X1JXT47GA]] of $G$.\nThen $H$ is solvable.\n\n## Remarks\n\nA proof uses the [[derived series|0Q5G5DJ4P2]]: $H^{(i)} \\subseteq G^{(i)}$ for each $i$, so $G^{(n)} = \\{e\\}$ implies $H^{(n)} = \\{e\\}$.\nMore generally, if there is an injective homomorphism from $H$ into a solvable group, then $H$ is solvable.\nFor example, each subgroup of $S_4$ is solvable.\n\n## Explanations\n\n### L1 human\n\nCommutators of elements of $H$ are also commutators in $G$.\nSo the derived series of $H$ stays inside the derived series of $G$.\nIf the larger series reaches the identity, the smaller one does too.\n",
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