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  "body": "\n## Statement\n\nLet $G$ be a [[nilpotent group|0Q7MMQHC1P]].\nThen $G$ is [[solvable|0FXJ7YFMMX]].\n\n## Remarks\n\nA central series $\\{e\\} = N_0 \\subseteq \\cdots \\subseteq N_n = G$ is a subnormal series, and each factor $N_{i+1}/N_i$ is central in $G/N_i$, so it is abelian.\nThe converse is false: $S_3$ is solvable, but it is not nilpotent.\nSo there are strict inclusions: abelian groups, then nilpotent groups, then solvable groups.\n\n## Explanations\n\n### L1 human\n\nA central series has layers that commute with the whole group.\nIn particular, each layer is abelian.\nSo a nilpotent group is solvable.\n",
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