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  "body": "\n## Definition\n\nLet a [[group|G5S1M9]] $G$ [[act|0WCY42EV9V]] on a nonempty set $X$.\nThe action is transitive when for all $x, y \\in X$ there is a $g \\in G$ with $g \\cdot x = y$.\n\n## Remarks\n\nThe action is transitive if and only if it has exactly one [[orbit|0YQPZB5Q1G]].\nThe action of $S_n$ on $\\{1, \\dots, n\\}$ is transitive.\nThe action of $G$ on the left cosets of a subgroup $H$ is transitive. Each transitive action of $G$ is isomorphic to such an action.\n\n## Explanations\n\n### L1 human\n\nA transitive action can move any point to any other point.\nSo the set does not split into separate pieces under the action.\nThe rotations of a square act transitively on its four corners.\n",
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