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  "body": "\n## Description\n\nLet $n \\ge 1$.\nThe *orthogonal group* is the set\n$$O(n) = \\{Q \\in M_n(\\R) : Q^T Q = I_n\\}$$\nof real $n \\times n$ [[orthogonal matrices|0W02QESPJQ]], with matrix multiplication.\nIt is a subgroup of the general linear group $GL_n(\\R)$.\n\n## Remarks\n\nIt is the group of linear isometries of $\\R^n$ with the dot product.\nEach element has determinant $1$ or $-1$, and $\\det \\colon O(n) \\to \\{1, -1\\}$ is a surjective homomorphism.\nIts kernel is the [[special orthogonal group|0CGYNDCWRR]].\n$O(n)$ is a compact Lie group of dimension $n(n-1)/2$.\n\n## Explanations\n\n### L1 human\n\nThe orthogonal group collects all rotations and reflections of $n$-dimensional space that fix the origin.\nComposing two of them gives another one, and each one can be undone.\nIn the plane, it consists of the rotations about the origin and the reflections in lines through the origin.\n",
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