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  "body": "\n## Definition\n\nLet a [[group|G5S1M9]] $G$ [[act|0WCY42EV9V]] on a set $X$, and let $x \\in X$.\nThe stabiliser of $x$ is the set $G_x = \\{g \\in G : g \\cdot x = x\\}$.\n\n## Remarks\n\nThe stabiliser $G_x$ is a [[subgroup|0X1JXT47GA]] of $G$, but in general it is not normal.\nFor the action of $S_n$ on $\\{1, \\dots, n\\}$, the stabiliser of $n$ is a subgroup isomorphic to $S_{n-1}$.\nFor the action of $G$ on itself by conjugation, the stabiliser of $a$ is the [[centraliser|02YYV36BFM]] $C_G(a)$.\n\n## Explanations\n\n### L1 human\n\nThe stabiliser of a point collects the group elements that keep this point in place.\nThey may move other points.\nFor the rotations of a square about its centre, only the identity keeps a corner in place.\n",
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