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  "body": "\n## Statement\n\nLet a [[group|G5S1M9]] $G$ [[act|0WCY42EV9V]] on a set $X$, and let $x \\in X$.\nThen the map $gG_x \\mapsto g \\cdot x$ is a well defined bijection from the set of the [[left cosets|03DEH43F4H]] of the [[stabiliser|0AYMP41THN]] $G_x$ to the [[orbit|0YQPZB5Q1G]] $G \\cdot x$.\nIn particular, $|G \\cdot x| = [G : G_x]$.\n\n## Remarks\n\nThe key equivalence is $g \\cdot x = h \\cdot x$ if and only if $g^{-1}h \\in G_x$, if and only if $gG_x = hG_x$.\nThe bijection is compatible with the actions: it sends $k \\cdot gG_x$ to $k \\cdot (g \\cdot x)$ for each $k \\in G$.\nSo each transitive action of $G$ is isomorphic to the [[action on the left cosets|0AZG100DKH]] of a stabiliser.\n\n## Explanations\n\n### L1 human\n\nTwo group elements move $x$ to the same point exactly when they differ by an element that fixes $x$.\nSo the points of the orbit correspond to the cosets of the stabiliser.\nA big stabiliser means a small orbit, and a small stabiliser means a big orbit.\n",
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  "name": "Orbit-stabiliser theorem",
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