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  "body": "\n## Statement\n\nLet a finite [[group|G5S1M9]] $G$ [[act|0WCY42EV9V]] on a set $X$, and let $x \\in X$.\nThen $|G \\cdot x| \\cdot |G_x| = |G|$.\nIn particular, the size of each [[orbit|0YQPZB5Q1G]] divides $|G|$.\n\n## Remarks\n\nThis follows from the [[orbit-stabiliser theorem|0E2HSCKZ6C]] and from $[G : G_x] \\cdot |G_x| = |G|$, which is [[Lagrange's theorem|3C53TD]].\nFor example, the rotation group of a cube acts transitively on the $6$ faces, and the stabiliser of a face has $4$ elements. So the group has order $24$.\n\n## Explanations\n\n### L1 human\n\nEach point of the orbit is reached by exactly $|G_x|$ group elements.\nSo the group elements are counted by the orbit size times the stabiliser size.\nThis is a good way to compute the order of a symmetry group.\n",
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  "msc": [
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  "name": "Size of an orbit times the size of the stabiliser",
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