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  "body": "\n## Statement\n\nLet $p$ be a prime number and let $G$ be a finite [[p-group|06QYRTX9KQ]].\nThen $G$ is [[nilpotent|0Q7MMQHC1P]].\n\n## Remarks\n\nEach quotient $G/Z_i$ of the [[upper central series|0XW54DH28V]] is a $p$-group, so it has a [[nontrivial centre|0RZCTT30H5]] when it is not trivial.\nSo the upper central series grows strictly until it reaches $G$, and $G$ is finite.\nFor $|G| = p^n$ with $n \\geq 2$, the nilpotency class of $G$ is at most $n - 1$.\n\n## Explanations\n\n### L1 human\n\nA nontrivial finite $p$-group always has a nontrivial centre.\nDivide by it, and the quotient is again a $p$-group with a nontrivial centre.\nRepeat this, and you climb the upper central series to the whole group.\n",
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