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  "body": "\n## Statement\n\nLet $p$ be a prime number, let $G$ be a finite [[p-group|06QYRTX9KQ]], and let $G$ [[act|0WCY42EV9V]] on a finite set $X$.\nLet $X^G$ be the set of the [[fixed points|0RRWNMYRXC]] of the action.\nThen $|X| \\equiv |X^G| \\pmod p$.\n\n## Remarks\n\nThe [[orbits|0ZC7FYE0NZ]] partition $X$, and the orbits with one point are exactly the fixed points.\nEach other orbit has a size that divides $|G|$ and is greater than $1$, so its size is divisible by $p$.\nA consequence: if $p$ does not divide $|X|$, then the action has at least one fixed point.\n\n## Explanations\n\n### L1 human\n\nThe orbits of a $p$-group have sizes that are powers of $p$.\nThe orbits of size $1$ are the fixed points, and all other orbits have a size divisible by $p$.\nSo counting modulo $p$ sees only the fixed points.\n",
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