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  "body": "\n## Statement\n\nLet $G$ be a [[group|G5S1M9]] and let $N$ be a [[normal subgroup|0CB0DX5BVV]] of $G$.\nIf $N$ and $G/N$ are [[solvable|0FXJ7YFMMX]], then $G$ is solvable.\n\n## Remarks\n\nIf $(G/N)^{(k)}$ is trivial, then $G^{(k)} \\subseteq N$, so $G^{(k+m)} \\subseteq N^{(m)}$ for each $m$.\nChoose $m$ with $N^{(m)} = \\{e\\}$; then $G^{(k+m)} = \\{e\\}$.\nWith [[subgroups|05Y0F9M0TS]] and [[quotients|027ZQS04SJ]], this gives: $G$ is solvable if and only if $N$ and $G/N$ are solvable.\nIn particular, direct products and semidirect products of solvable groups are solvable.\n\n## Explanations\n\n### L1 human\n\nIf the bottom part and the top part of a group can both be built from abelian layers, then so can the whole group.\nYou stack the layers of the top part on the layers of the bottom part.\nFor example, $S_4$ is built from $V$, $A_4/V$ and $S_4/A_4$.\n",
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