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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]] with identity $e$, and let $X$ be a set.\nA (left) action of $G$ on $X$ is a map $G \\times X \\to X$, $(g, x) \\mapsto g \\cdot x$, with these two properties:\n\n1. $e \\cdot x = x$ for each $x \\in X$.\n2. $g \\cdot (h \\cdot x) = (gh) \\cdot x$ for all $g, h \\in G$ and $x \\in X$.\n\n## Remarks\n\nFor each $g \\in G$, the map $x \\mapsto g \\cdot x$ is a bijection of $X$, with inverse $x \\mapsto g^{-1} \\cdot x$.\nThe map that sends $g$ to this bijection is a [[group homomorphism|0JYTG5K9Y4]] from $G$ to the [[symmetric group|0515A6D2Y1]] $\\operatorname{Sym}(X)$.\nConversely, each homomorphism $G \\to \\operatorname{Sym}(X)$ gives an action of $G$ on $X$.\nA right action is a map $X \\times G \\to X$ with $x \\cdot e = x$ and $(x \\cdot g) \\cdot h = x \\cdot (gh)$.\n\n## Explanations\n\n### L1 human\n\nA group action lets the elements of a group move the points of a set.\nThe identity does not move any point, and moving by $h$ and then by $g$ is the same as moving by $gh$.\nFor example, the rotations of a square move the four corners of the square.\n",
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