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text: '[summary: Given a [-3gd group] $G$ [-3t9 acting] on a [-3jz set] $X$, the [-4mz stabiliser] of some [-element_mathematics element] $x \\in X$ is a [-subgroup subgroup] of $G$.\n]\n\nLet $G$ be a [-3gd] which [3t9 acts] on the [-3jz set] $X$.\nThen for every $x \\in X$, the [4mz stabiliser] $\\mathrm{Stab}_G(x)$ is a [-subgroup subgroup] of $G$.\n\n# Proof\n\nWe must check the group axioms.\n\n- The [-54p identity], $e$, is in the stabiliser because $e(x) = x$; this is part of the definition of a group action.\n- [-3gy Closure] is satisfied: if $g(x) = x$ and $h(x) = x$, then $(gh)(x) = g(h(x))$ by definition of a group action, but that is $g(x) = x$.\n- [-3h4 Associativity] is inherited from the parent group.\n- [-inverse_mathematics Inverses]: if $g(x) = x$ then $g^{-1}(x) = g^{-1} g(x) = e(x) = x$.',
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