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  "body": "\n## Definition\n\nLet a [[group|G5S1M9]] $G$ [[act|0WCY42EV9V]] on a set $X$.\nThe action is faithful when, for each $g \\in G$, $g \\cdot x = x$ for all $x \\in X$ implies $g = e$.\n\n## Remarks\n\nThe action is faithful if and only if the associated homomorphism $G \\to \\operatorname{Sym}(X)$ is injective.\nSo a faithful action identifies $G$ with a subgroup of $\\operatorname{Sym}(X)$.\nThe action of $G$ on itself by conjugation is faithful if and only if the [[centre|0DZ93493VN]] of $G$ is trivial.\n\n## Explanations\n\n### L1 human\n\nIn a faithful action, each element of the group that is not the identity moves at least one point.\nSo you can tell the group elements apart by watching how they move the points.\nThe group is then a group of permutations of the set.\n",
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