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text: '[summary: \nA [-3t9 group action] of a group $G$ acting on a set $X$ describes how $G$ sends elements of $X$ to other elements of $X$. Given a specific element $x \\in X$, the [-4mz stabiliser] is all those elements of the group which send $x$ back to itself, and the [-4v8 orbit] of $x$ is all the elements to which $x$ can get sent. \n\nThis theorem tells you that $G$ is divided into equal-sized pieces using $x$. Each piece "looks like" the stabilizer of $x$ (and is the same size), and the orbit of $x$ tells you how to "move the piece around" over $G$ in order to cover it. \n\nPut another way, each element $y$ in the orbit of $x$ is transformed "in the same way" by $G$ relative to $y$. \n\nThis theorem is closely related to [-4jn Lagrange's Theorem]. \n]\n\n[summary(Technical): Let $G$ be a finite [-3gd], [3t9 acting] on a set $X$. Let $x \\in X$.\nWriting $\\mathrm{Stab}_G(x)$ for the [4mz stabiliser] of $x$, and $\\mathrm{Orb}_G(x)$ for the [4v8 orbit] of $x$, we have $$|G| = |\\mathrm{Stab}_G(x)| \\times |\\mathrm{Orb}_G(x)|$$ where $| \\cdot |$ refers to the size of a set.]\n\nLet $G$ be a finite [-3gd], [3t9 acting] on a set $X$. Let $x \\in X$.\nWriting $\\mathrm{Stab}_G(x)$ for the [4mz stabiliser] of $x$, and $\\mathrm{Orb}_G(x)$ for the [4v8 orbit] of $x$, we have $$|G| = |\\mathrm{Stab}_G(x)| \\times |\\mathrm{Orb}_G(x)|$$ where $| \\cdot |$ refers to the size of a set.\n\nThis statement generalises to infinite groups, where the same proof goes through to show that there is a [499 bijection] between the [4j4 left cosets] of the group $\\mathrm{Stab}_G(x)$ and the orbit $\\mathrm{Orb}_G(x)$.\n\n# Proof\n\nRecall that the [-4lt] of the parent group.\n\nFirstly, it is enough to show that there is a bijection between the left cosets of the stabiliser, and the orbit.\nIndeed, then $$|\\mathrm{Orb}_G(x)| |\\mathrm{Stab}_G(x)| = |\\{ \\text{left cosets of} \\ \\mathrm{Stab}_G(x) \\}| |\\mathrm{Stab}_G(x)|$$\nbut the right-hand side is simply $|G|$ because an element of $G$ is specified exactly by specifying an element of the stabiliser and a coset.\n(This follows because the [4j5 cosets partition the group].)\n\n## Finding the bijection\n\nDefine $\\theta: \\mathrm{Orb}_G(x) \\to \\{ \\text{left cosets of} \\ \\mathrm{Stab}_G(x) \\}$, by $$g(x) \\mapsto g \\mathrm{Stab}_G(x)$$\n\nThis map is well-defined: note that any element of $\\mathrm{Orb}_G(x)$ is given by $g(x)$ for some $g \\in G$, so we need to show that if $g(x) = h(x)$, then $g \\mathrm{Stab}_G(x) = h \\mathrm{Stab}_G(x)$.\nThis follows: $h^{-1}g(x) = x$ so $h^{-1}g \\in \\mathrm{Stab}_G(x)$.\n\nThe map is [4b7 injective]: if $g \\mathrm{Stab}_G(x) = h \\mathrm{Stab}_G(x)$ then we need $g(x)=h(x)$.\nBut this is true: $h^{-1} g \\in \\mathrm{Stab}_G(x)$ and so $h^{-1}g(x) = x$, from which $g(x) = h(x)$.\n\nThe map is [4bg surjective]: let $g \\mathrm{Stab}_G(x)$ be a left coset.\nThen $g(x) \\in \\mathrm{Orb}_G(x)$ by definition of the orbit, so $g(x)$ gets taken to $g \\mathrm{Stab}_G(x)$ as required.\n\nHence $\\theta$ is a well-defined bijection.',
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