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  "body": "\n## Summary\n\nA Coxeter group is a [[group|G5S1M9]] generated by elements of order $2$.\nThe only relations prescribe the order of the product of each pair of generators.\nThe finite Coxeter groups are exactly the finite groups generated by [[reflections|06BRDF3F0E]] of a Euclidean space, such as the [[symmetric groups|0515A6D2Y1]] and the [[dihedral groups|0JZ27DMJ5D]].\nInfinite Coxeter groups include the affine Weyl groups and groups generated by reflections in the sides of hyperbolic polygons whose angles have the form $\\pi / m$.\nThey appear in Lie theory, in the theory of buildings and in geometric group theory.\n\n## Explanations\n\n### L1 human\n\nPut two mirrors at an angle of $\\pi / m$.\nEach reflection undoes itself, so it has order $2$, and the product of the two reflections is a rotation by $2\\pi / m$, of order $m$.\nThe group generated by the two reflections is the dihedral group of order $2m$, the symmetries of a regular $m$-gon.\nA Coxeter group is any group built in the same way from several such mirrors, where only these pairwise rules are imposed.\n",
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