"automorphism group" meaning in All languages combined

{
  "forms": [
    {
      "form": "automorphism groups",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
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      "args": {},
      "expansion": "automorphism group (plural automorphism groups)",
      "name": "en-noun"
    }
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        {
          "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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          "langcode": "en",
          "name": "Group theory",
          "orig": "en:Group theory",
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            "Mathematics",
            "Formal sciences",
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      ],
      "examples": [
        {
          "text": "The automorphism group of a set is the set's symmetric group.",
          "type": "example"
        },
        {
          "text": "The automorphism group of a finite-dimensional vector space (respectively, projective space) is the space's general linear group (respectively, projective linear group).",
          "type": "example"
        },
        {
          "ref": "1984, José M. Isidro, László L. Stachó, Holomorphic Automorphism Groups in Banach Spaces: An Elementary Introduction, Elsevier (North-Holland), page vi",
          "text": "Moreover, we think that this approach to the automorphism groups of Banach space domains may also serve as motivating and illustrative material in introducing students to the theory of Lie groups and complex manifolds.",
          "type": "quotation"
        },
        {
          "ref": "1993, Matatyahu Rubin, The Reconstruction of Trees from Their Automorphism Groups, American Mathematical Society, page 5",
          "text": "Recently Droste, Holland and Macpherson in [DHM1], [DHM2] and [DHM3] investigated various questions concerning the automorphism groups of homogeneous #x5C;aleph#x5F;0-categorical trees.",
          "type": "quotation"
        },
        {
          "ref": "2002, Masamichi Takesaki, Theory of Operator Algebras II, Springer, page 91",
          "text": "Unlike the classical integration theory, the non-commutative theory comes with the intrinsic dynamics of the modular automorphism group. This chapter is devoted to the elementary theory of the modular automorphism groups.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "The group of automorphisms of a given set or other mathematical object, with the group operation being function composition."
      ],
      "hyponyms": [
        {
          "word": "Galois group"
        },
        {
          "word": "general linear group"
        },
        {
          "word": "projective linear group"
        }
      ],
      "id": "en-automorphism_group-en-noun-oko4wypR",
      "links": [
        [
          "group theory",
          "group theory"
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        [
          "group",
          "group"
        ],
        [
          "automorphism",
          "automorphism"
        ],
        [
          "set",
          "set"
        ],
        [
          "operation",
          "operation"
        ],
        [
          "function composition",
          "function composition"
        ]
      ],
      "raw_glosses": [
        "(group theory) The group of automorphisms of a given set or other mathematical object, with the group operation being function composition."
      ],
      "related": [
        {
          "english": "type of automorphism",
          "word": "group automorphism"
        }
      ],
      "synonyms": [
        {
          "english": "chiefly in geometry",
          "sense": "symmetric group",
          "word": "symmetry group"
        }
      ],
      "topics": [
        "group-theory",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "automorphism group"
      ]
    }
  ],
  "word": "automorphism group"
}

{
  "forms": [
    {
      "form": "automorphism groups",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "automorphism group (plural automorphism groups)",
      "name": "en-noun"
    }
  ],
  "hyponyms": [
    {
      "word": "Galois group"
    },
    {
      "word": "general linear group"
    },
    {
      "word": "projective linear group"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "english": "type of automorphism",
      "word": "group automorphism"
    }
  ],
  "senses": [
    {
      "categories": [
        "English countable nouns",
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        "English multiword terms",
        "English nouns",
        "English terms with quotations",
        "English terms with usage examples",
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      ],
      "examples": [
        {
          "text": "The automorphism group of a set is the set's symmetric group.",
          "type": "example"
        },
        {
          "text": "The automorphism group of a finite-dimensional vector space (respectively, projective space) is the space's general linear group (respectively, projective linear group).",
          "type": "example"
        },
        {
          "ref": "1984, José M. Isidro, László L. Stachó, Holomorphic Automorphism Groups in Banach Spaces: An Elementary Introduction, Elsevier (North-Holland), page vi",
          "text": "Moreover, we think that this approach to the automorphism groups of Banach space domains may also serve as motivating and illustrative material in introducing students to the theory of Lie groups and complex manifolds.",
          "type": "quotation"
        },
        {
          "ref": "1993, Matatyahu Rubin, The Reconstruction of Trees from Their Automorphism Groups, American Mathematical Society, page 5",
          "text": "Recently Droste, Holland and Macpherson in [DHM1], [DHM2] and [DHM3] investigated various questions concerning the automorphism groups of homogeneous #x5C;aleph#x5F;0-categorical trees.",
          "type": "quotation"
        },
        {
          "ref": "2002, Masamichi Takesaki, Theory of Operator Algebras II, Springer, page 91",
          "text": "Unlike the classical integration theory, the non-commutative theory comes with the intrinsic dynamics of the modular automorphism group. This chapter is devoted to the elementary theory of the modular automorphism groups.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "The group of automorphisms of a given set or other mathematical object, with the group operation being function composition."
      ],
      "links": [
        [
          "group theory",
          "group theory"
        ],
        [
          "group",
          "group"
        ],
        [
          "automorphism",
          "automorphism"
        ],
        [
          "set",
          "set"
        ],
        [
          "operation",
          "operation"
        ],
        [
          "function composition",
          "function composition"
        ]
      ],
      "raw_glosses": [
        "(group theory) The group of automorphisms of a given set or other mathematical object, with the group operation being function composition."
      ],
      "topics": [
        "group-theory",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "automorphism group"
      ]
    }
  ],
  "synonyms": [
    {
      "english": "chiefly in geometry",
      "sense": "symmetric group",
      "word": "symmetry group"
    }
  ],
  "word": "automorphism group"
}