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text: 'An action of a [-3gd] $G$ on a [-3jz] $X$ is a function $\\alpha : G \\times X \\to X$ ([3vl colon-to notation]), which is often written $(g, x) \\mapsto gx$ ([3vm mapsto notation]), with $\\alpha$ omitted from the notation, such that\n\n1. $ex = x$ for all $x \\in X$, where $e$ is the identity, and\n2. $g(hx) = (gh)x$ for all $g, h \\in G, x \\in X$, where $gh$ implicitly refers to the group operation in $G$ (also omitted from the notation).\n\nEquivalently, via [currying], an action of $G$ on $X$ is a [47t group homomorphism] $G \\to \\text{Aut}(X)$, where $\\text{Aut}(X)$ is the [automorphism_group automorphism group] of $X$ (so for sets, the group of all bijections $X \\to X$, but phrasing the definition this way makes it natural to generalize to other [category_theory categories]). It's a good exercise to verify this; Arbital [49c has a proof].\n\nGroup actions are used to make precise the notion of "symmetry" in mathematics. \n\n# Examples\n\nLet $X = \\mathbb{R}^2$ be the [Euclidean_geometry Euclidean plane]. There's a group acting on $\\mathbb{R}^2$ called the [Euclidean_group Euclidean group] $ISO(2)$ which consists of all functions $f : \\mathbb{R}^2 \\to \\mathbb{R}^2$ preserving distances between two points (or equivalently all [isometry isometries]). Its elements include translations, rotations about various points, and reflections about various lines. ',
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