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  "body": "\n## Example\n\nLet $G$ be a [[group|G5S1M9]].\nThe map $G \\times G \\to G$, $(g, x) \\mapsto gxg^{-1}$, is an [[action|0WCY42EV9V]] of $G$ on the set $G$.\nIndeed, $exe^{-1} = x$ and $g(hxh^{-1})g^{-1} = (gh)x(gh)^{-1}$.\nThe [[orbits|0YQPZB5Q1G]] of this action are the [[conjugacy classes|0DEBXAJ5SA]] of $G$.\nThe [[stabiliser|0AYMP41THN]] of $x$ is the [[centraliser|02YYV36BFM]] $C_G(x)$.\nThe [[fixed points|0RRWNMYRXC]] are the elements of the [[centre|0DZ93493VN]] $Z(G)$.\nFor an [[abelian group|082GHNPJ53]], each element is a fixed point, and each orbit has one element.\n",
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        "locator": "Conjugation as a group action, centralizers, and the class equation",
        "url": "https://en.wikipedia.org/wiki/Conjugacy_class"
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  "name": "Action by conjugation",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/0P8/0P8CY9V437.md",
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