"orthogonal group" meaning in All languages combined

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  "lang_code": "en",
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      "derived": [
        {
          "word": "indefinite orthogonal group"
        },
        {
          "word": "special orthogonal group"
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        {
          "ref": "1998, Robert L. Griess, Jr., Twelve Sporadic Groups, Springer, page 4:",
          "text": "The symbol O^ε(n,q) for orthogonal groups has been well established in finite group theory as and, throughout the mathematics community, O(n, K) stands for an orthogonal group when K is the real or complex field.",
          "type": "quote"
        },
        {
          "ref": "1999, Gunter Malle, B.H. Matzat, Inverse Galois Theory, Springer, page 146:",
          "text": "Theorem 7.4. Let n ≥ 1. For odd primes p#x5C;not#x5C;equiv#x5C;pm 1#x5C;pmod#x7B;24#x7D; the odd-dimensional orthogonal groups O#x5F;#x7B;2n#x2B;1#x7D;(p) possess GA-realizations over #x5C;mathbb#x7B;Q#x7D;.",
          "type": "quote"
        },
        {
          "ref": "2007, Marcelo Epstein, Marek Elzanowski, Material Inhomogeneities and their Evolution: A Geometric Approach, Springer, page 106:",
          "text": "The normalizer of the full orthogonal group within the general linear group can be shown to consist of all (commutative) products of spherical dilatations and orthogonal transformations.",
          "type": "quote"
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        "For given n and field F (especially where F is the real numbers), the group of n × n orthogonal matrices with elements in F, where the group operation is matrix multiplication."
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      "id": "en-orthogonal_group-en-noun-hkoM8Vy3",
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        "(group theory) For given n and field F (especially where F is the real numbers), the group of n × n orthogonal matrices with elements in F, where the group operation is matrix multiplication."
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      "related": [
        {
          "word": "symplectic group"
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          "word": "unitary group"
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  "word": "orthogonal group"
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      "word": "indefinite orthogonal group"
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      "word": "special orthogonal group"
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    {
      "word": "symplectic group"
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          "ref": "1998, Robert L. Griess, Jr., Twelve Sporadic Groups, Springer, page 4:",
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          "type": "quote"
        },
        {
          "ref": "1999, Gunter Malle, B.H. Matzat, Inverse Galois Theory, Springer, page 146:",
          "text": "Theorem 7.4. Let n ≥ 1. For odd primes p#x5C;not#x5C;equiv#x5C;pm 1#x5C;pmod#x7B;24#x7D; the odd-dimensional orthogonal groups O#x5F;#x7B;2n#x2B;1#x7D;(p) possess GA-realizations over #x5C;mathbb#x7B;Q#x7D;.",
          "type": "quote"
        },
        {
          "ref": "2007, Marcelo Epstein, Marek Elzanowski, Material Inhomogeneities and their Evolution: A Geometric Approach, Springer, page 106:",
          "text": "The normalizer of the full orthogonal group within the general linear group can be shown to consist of all (commutative) products of spherical dilatations and orthogonal transformations.",
          "type": "quote"
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      ],
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        "For given n and field F (especially where F is the real numbers), the group of n × n orthogonal matrices with elements in F, where the group operation is matrix multiplication."
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        "(group theory) For given n and field F (especially where F is the real numbers), the group of n × n orthogonal matrices with elements in F, where the group operation is matrix multiplication."
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