"orthogonal group" meaning in English

{
  "forms": [
    {
      "form": "orthogonal groups",
      "tags": [
        "plural"
      ]
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  ],
  "head_templates": [
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      "args": {},
      "expansion": "orthogonal group (plural orthogonal groups)",
      "name": "en-noun"
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
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          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
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          "source": "w"
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        {
          "kind": "topical",
          "langcode": "en",
          "name": "Group theory",
          "orig": "en:Group theory",
          "parents": [
            "Algebra",
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
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        }
      ],
      "derived": [
        {
          "word": "indefinite orthogonal group"
        },
        {
          "word": "special orthogonal group"
        }
      ],
      "examples": [
        {
          "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": "quotation"
        },
        {
          "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": "quotation"
        },
        {
          "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": "quotation"
        }
      ],
      "glosses": [
        "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."
      ],
      "id": "en-orthogonal_group-en-noun-hkoM8Vy3",
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          "group theory",
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          "field",
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        [
          "real numbers",
          "real numbers"
        ],
        [
          "orthogonal matrices",
          "orthogonal matrix"
        ]
      ],
      "raw_glosses": [
        "(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."
      ],
      "related": [
        {
          "word": "symplectic group"
        },
        {
          "word": "unitary group"
        }
      ],
      "topics": [
        "group-theory",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "orthogonal group"
      ]
    }
  ],
  "word": "orthogonal group"
}

{
  "derived": [
    {
      "word": "indefinite orthogonal group"
    },
    {
      "word": "special orthogonal group"
    }
  ],
  "forms": [
    {
      "form": "orthogonal groups",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
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      "expansion": "orthogonal group (plural orthogonal groups)",
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "symplectic group"
    },
    {
      "word": "unitary group"
    }
  ],
  "senses": [
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      "categories": [
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        "en:Group theory"
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      "examples": [
        {
          "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": "quotation"
        },
        {
          "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": "quotation"
        },
        {
          "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": "quotation"
        }
      ],
      "glosses": [
        "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."
      ],
      "links": [
        [
          "group theory",
          "group theory"
        ],
        [
          "field",
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        ],
        [
          "real numbers",
          "real numbers"
        ],
        [
          "orthogonal matrices",
          "orthogonal matrix"
        ]
      ],
      "raw_glosses": [
        "(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."
      ],
      "topics": [
        "group-theory",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "orthogonal group"
      ]
    }
  ],
  "word": "orthogonal group"
}