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  "body": "\n## Summary\n\nThe classical groups are the [[general linear groups|0FCHE902D7]] and [[special linear groups|0V9PMN0RBH]], and the groups of linear maps that preserve a nondegenerate form.\nFor symmetric, alternating and Hermitian forms these are the [[orthogonal|0TKYAEJ8N7]], symplectic and [[unitary groups|0R9R0Y1HDK]].\nThe name goes back to Hermann Weyl's book on the classical groups.\nOver $\\C$ they give all the complex simple Lie groups except five exceptional ones, up to local isomorphism.\nOver finite fields, their projective versions give several infinite families of finite simple groups.\n\n## Explanations\n\n### L1 human\n\nA classical group is a group of matrices defined by one natural condition.\nThe general linear group keeps only invertibility, and the special linear group asks for determinant $1$.\nThe orthogonal group keeps lengths and angles, as rotations and reflections of space do.\nThe symplectic group keeps an area-like form, which appears in classical mechanics, and the unitary group keeps the length of complex vectors, as in quantum mechanics.\nFor example, the rotations of three-dimensional space form the special orthogonal group $\\mathrm{SO}(3)$.\n",
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