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  text: '[summary: The stabilizer of an element $x$ under the action of a group $G$ is the set ([4lt actually a subgroup]) of elements of $G$ which leave $x$ unchanged. ]\n\nLet the [-3gd] $G$ [3t9 act] on the set $X$.\nThen for each element $x \\in X$, the *stabiliser* of $x$ under $G$ is $\\mathrm{Stab}_G(x) = \\{ g \\in G: g(x) = x \\}$.\nThat is, it is the collection of elements of $G$ which do not move $x$ under the action.\n\nThe stabiliser of $x$ is a subgroup of $G$, for any $x \\in X$. ([4lt Proof.])\n\nA closely related notion is that of the [4v8 orbit] of $x$, and the very important [4l8 Orbit-Stabiliser theorem] linking the two.',
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