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  "body": "\n## Proof\n\nWe combine the orbit-stabiliser theorem with the index formula.\n\n1. Let $G$ be finite, acting on $X$, and let $x \\in X$. The [[stabiliser|0RGGF8ZS6D]] $G_x$ is a subgroup of $G$. Indeed, $e \\cdot x = x$ gives $e \\in G_x$, and $a, b \\in G_x$ gives $(a b^{-1}) \\cdot x = a \\cdot (b^{-1} \\cdot (b \\cdot x)) = x$.\n2. By the [[orbit-stabiliser theorem|0E2HSCKZ6C]], $\\abs{G \\cdot x} = [G : G_x]$.\n3. Since $G$ is finite and $G_x$ is a subgroup, [[$\\abs{G} = [G : G_x] \\cdot \\abs{G_x}$|04DC0SRN2K]].\n4. Substituting step 2 into step 3 gives $\\abs{G \\cdot x} \\cdot \\abs{G_x} = \\abs{G}$.\n5. Since $\\abs{G_x}$ is a positive integer, step 4 shows that $\\abs{G \\cdot x}$ divides $\\abs{G}$. So the size of each [[orbit|00F2JFA41J]] divides $\\abs{G}$.\n\n## Idea\n\nThe orbit is in bijection with the cosets of the stabiliser. The cosets all have the size of the stabiliser and fill up the group, so the two sizes multiply to $\\abs{G}$.\n",
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