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  "body": "\n## Description\n\nThe Euclidean group $\\mathrm{E}(n)$ is the set of all [[Euclidean isometries|0J1N3WTKM4]] of $\\mathbb{R}^n$, with the composition of maps as the operation. It is a [[group|G5S1M9]]: the identity map is the neutral element, and the inverse of an isometry is an isometry.\nEach element has the form $x \\mapsto A x + b$, with $A$ in the orthogonal group $\\mathrm{O}(n)$ and $b \\in \\mathbb{R}^n$.\nThe direct isometries, with $\\det A = 1$, form a subgroup $\\mathrm{SE}(n)$ of index $2$. Its elements are the rigid motions.\n\n## Remarks\n\nThe [[translations|087VJM5Q77]] form an abelian normal subgroup $T \\cong \\mathbb{R}^n$, and $\\mathrm{E}(n) / T \\cong \\mathrm{O}(n)$.\nThe group $\\mathrm{E}(n)$ has $n(n+1)/2$ degrees of freedom: $3$ for $n = 2$ and $6$ for $n = 3$.\nThe group is not abelian for $n \\geq 2$. For example, a rotation about the origin and a translation do not commute in general.\nThe symmetry group of each figure in $\\mathbb{R}^n$ is a subgroup of $\\mathrm{E}(n)$.\n\n## Explanations\n\n### L1 human\n\nThe Euclidean group collects all the ways to move space without changing distances. Two such moves done one after the other give again such a move, and each move can be undone. In the plane, three numbers fix a direct move: two for a shift and one for a turn.\n",
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