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  "body": "\n## Definition\n\nLet a [[group|G5S1M9]] $G$ [[act|0WCY42EV9V]] on a set $X$.\nA point $x \\in X$ is a fixed point of the action when $g \\cdot x = x$ for each $g \\in G$.\nThe set of all fixed points is written $X^G$.\nFor $g \\in G$, the set of points fixed by $g$ is $X^g = \\{x \\in X : g \\cdot x = x\\}$.\n\n## Remarks\n\nA point $x$ is a fixed point if and only if its [[orbit|0YQPZB5Q1G]] is $\\{x\\}$, if and only if its [[stabiliser|0AYMP41THN]] is $G$.\nFor the action of $G$ on itself by conjugation, the fixed points are the elements of the [[centre|0DZ93493VN]] $Z(G)$.\nThe sets $X^g$ appear in [[Burnside's lemma|0Z47M1B2VS]].\n\n## Explanations\n\n### L1 human\n\nA fixed point is a point that no element of the group moves.\nFor the rotations of a square about its centre, the centre is a fixed point.\nThe corners are not fixed points, because a quarter turn moves them.\n",
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