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  "body": "\n## Statement\n\nLet $G$ be a [[solvable group|0FXJ7YFMMX]] and let $\\varphi \\colon G \\to K$ be a surjective [[group homomorphism|0JYTG5K9Y4]].\nThen $K$ is solvable.\nIn particular, $G/N$ is solvable for each [[normal subgroup|0CB0DX5BVV]] $N$ of $G$.\n\n## Remarks\n\nA proof uses the [[derived series|0Q5G5DJ4P2]]: $\\varphi(G^{(i)}) = K^{(i)}$ for each $i$, because $\\varphi$ maps commutators onto commutators.\nSo $G^{(n)} = \\{e\\}$ implies $K^{(n)} = \\{e\\}$.\nThe second part follows with the projection $G \\to G/N$.\n\n## Explanations\n\n### L1 human\n\nA surjective homomorphism sends the commutators of $G$ onto the commutators of $K$.\nSo the derived series of $K$ is the image of the derived series of $G$.\nIf one reaches the identity, the image does too.\n",
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