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  "body": "\n## Statement\n\nLet a [[group|G5S1M9]] $G$ [[act|0WCY42EV9V]] on a set $X$.\nThe relation $x \\sim y$ if and only if $y = g \\cdot x$ for some $g \\in G$ is an equivalence relation on $X$.\nIts equivalence classes are the [[orbits|0YQPZB5Q1G]], so each point of $X$ lies in exactly one orbit.\n\n## Remarks\n\nReflexivity uses $e \\cdot x = x$, symmetry uses $x = g^{-1} \\cdot (g \\cdot x)$, and transitivity uses $h \\cdot (g \\cdot x) = (hg) \\cdot x$.\nIf $X$ is finite, then $|X|$ is the sum of the sizes of the distinct orbits.\nThe [[cosets|03DEH43F4H]] of a subgroup and the [[conjugacy classes|0DEBXAJ5SA]] of a group are examples of orbits.\n\n## Explanations\n\n### L1 human\n\nThe group action cuts the set into separate pieces, the orbits.\nInside one piece, each point can reach each other point.\nNo point can reach a point of another piece.\n",
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