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  "body": "\n## Statement\n\nLet $G$ be a finite [[group|G5S1M9]].\nThe following conditions are equivalent:\n\n1. $G$ is [[nilpotent|0Q7MMQHC1P]].\n2. For each prime number $p$, $G$ has exactly one [[Sylow p-subgroup|0053CQE9Y2]].\n3. $G$ is the [[internal direct product|0VBZH90R1A]] of its Sylow subgroups, one for each prime divisor of $|G|$.\n\n## Remarks\n\nThe direction from 3 to 1 holds because [[finite p-groups are nilpotent|081WTKYX6C]] and a finite direct product of nilpotent groups is nilpotent.\nThe direction from 1 to 2 uses the [[Frattini argument|0QTZDKJEDV]]. In a nilpotent group, a proper subgroup is strictly smaller than its normaliser.\nFor example, $\\mathbb{Z}/12\\mathbb{Z} \\cong \\mathbb{Z}/4\\mathbb{Z} \\times \\mathbb{Z}/3\\mathbb{Z}$ is nilpotent, but $S_3$ has three Sylow $2$-subgroups, so it is not nilpotent.\n\n## Explanations\n\n### L1 human\n\nA finite nilpotent group splits into its $p$-parts, one for each prime.\nThe parts do not interact, so the group is their direct product.\nConversely, a direct product of $p$-groups is nilpotent.\n",
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