{
  "aliases": [
    "orbit-counting-theorem",
    "cauchy-frobenius-lemma"
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  "body": "\n## Statement\n\nLet a finite [[group|G5S1M9]] $G$ [[act|0WCY42EV9V]] on a finite set $X$.\nFor $g \\in G$, let $X^g = \\{x \\in X : g \\cdot x = x\\}$.\nThen the number of [[orbits|0YQPZB5Q1G]] is $|X/G| = \\frac{1}{|G|} \\sum_{g \\in G} |X^g|$.\n\n## Remarks\n\nThe proof counts the pairs $(g, x)$ with $g \\cdot x = x$ in two ways, and uses $|G_x| = |G| / |G \\cdot x|$.\nFor example, the rotation group $\\mathbb{Z}/4\\mathbb{Z}$ of a square acts on the $2^4 = 16$ colourings of the corners with two colours.\nThe fixed colourings number $16$, $2$, $4$ and $2$, so there are $(16 + 2 + 4 + 2)/4 = 6$ colourings up to rotation.\n\n## History\n\nWilliam Burnside stated and proved the lemma in his 1897 book on finite groups, and he attributed it to Frobenius, 1887.\nCauchy already knew the formula in 1845.\nFor this reason, the lemma is sometimes called \"the lemma that is not Burnside's\".\n\n## Explanations\n\n### L1 human\n\nThe number of orbits is the average number of points that a group element fixes.\nThis lets you count objects up to symmetry without listing them.\nYou count the fixed objects for each symmetry, add, and divide by the number of symmetries.\n",
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