"Coxeter group" meaning in English

{
  "etymology_text": "Named for British-born Canadian mathematician H. S. M. Coxeter (1907–2003).",
  "forms": [
    {
      "form": "Coxeter groups",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "Coxeter group (plural Coxeter groups)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        {
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
            "Entry maintenance"
          ],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "English entries with language name categories using raw markup",
          "parents": [
            "Entries with language name categories using raw markup",
            "Entry maintenance"
          ],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "English terms with non-redundant non-automated sortkeys",
          "parents": [
            "Terms with non-redundant non-automated sortkeys",
            "Entry maintenance"
          ],
          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Geometry",
          "orig": "en:Geometry",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Group theory",
          "orig": "en:Group theory",
          "parents": [
            "Algebra",
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Mathematics",
          "orig": "en:Mathematics",
          "parents": [
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "examples": [
        {
          "ref": "1993, B. Mühlherr, “Coxeter groups in Coxeter groups”, in Albrecht Beutelspacher, F. de Clerck, editors, Finite Geometries and Combinatorics, page 277",
          "text": "We may ask in general which Coxeter groups arise as subgroups of a given Coxeter group. This question is of course far too general. However, there are Coxeter groups which arise canonically as subgroups of a given Coxeter group.",
          "type": "quotation"
        },
        {
          "ref": "2007, Michael B. Smythe, Julian Webster, “12: Discrete Spatial Models”, in Marco Aiello, Ian Pratt-Hartmann, Johan van Benthem, editors, Handbook of Spatial Logics, page 795",
          "text": "The above definitions, of Cayley graph, reflection, wall, half-space, folding, Bruhat order, etc., work for any Coxeter group.",
          "type": "quotation"
        },
        {
          "ref": "2012, Daniel Allcock, “The Reflective Lorentzian Lattices of Rank 3”, in Memoirs of the American Mathematical Society, number 1033, page vii",
          "text": "The problem of classifying all reflective lattices of given rank is also of interest in its own right from the perspectives of Coxeter groups and arithmetic subgroups of O(n,1).",
          "type": "quotation"
        }
      ],
      "glosses": [
        "Any of a class of groups whose finite cases are precisely the finite reflection groups (including the symmetry groups of polytopes), but which are more varied in their infinite cases, and whose range of application encompasses various areas of mathematics."
      ],
      "id": "en-Coxeter_group-en-noun-ZhZBsqiE",
      "links": [
        [
          "mathematics",
          "mathematics"
        ],
        [
          "geometry",
          "geometry"
        ],
        [
          "group theory",
          "group theory"
        ],
        [
          "class",
          "class"
        ],
        [
          "group",
          "group"
        ],
        [
          "finite",
          "finite"
        ],
        [
          "reflection group",
          "reflection group"
        ],
        [
          "symmetry group",
          "symmetry group"
        ],
        [
          "polytope",
          "polytope"
        ],
        [
          "application",
          "application"
        ]
      ],
      "raw_glosses": [
        "(mathematics, geometry, group theory) Any of a class of groups whose finite cases are precisely the finite reflection groups (including the symmetry groups of polytopes), but which are more varied in their infinite cases, and whose range of application encompasses various areas of mathematics."
      ],
      "related": [
        {
          "word": "Coxeter-Dynkin diagram"
        },
        {
          "word": "Coxeter element"
        },
        {
          "word": "Coxeter matrix"
        },
        {
          "word": "Coxeter number"
        },
        {
          "word": "Coxeter system"
        }
      ],
      "topics": [
        "geometry",
        "group-theory",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "Coxeter group",
        "Harold Scott MacDonald Coxeter"
      ]
    }
  ],
  "word": "Coxeter group"
}

{
  "etymology_text": "Named for British-born Canadian mathematician H. S. M. Coxeter (1907–2003).",
  "forms": [
    {
      "form": "Coxeter groups",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "Coxeter group (plural Coxeter groups)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "Coxeter-Dynkin diagram"
    },
    {
      "word": "Coxeter element"
    },
    {
      "word": "Coxeter matrix"
    },
    {
      "word": "Coxeter number"
    },
    {
      "word": "Coxeter system"
    }
  ],
  "senses": [
    {
      "categories": [
        "English countable nouns",
        "English entries with incorrect language header",
        "English entries with language name categories using raw markup",
        "English eponyms",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms with non-redundant non-automated sortkeys",
        "English terms with quotations",
        "en:Geometry",
        "en:Group theory",
        "en:Mathematics"
      ],
      "examples": [
        {
          "ref": "1993, B. Mühlherr, “Coxeter groups in Coxeter groups”, in Albrecht Beutelspacher, F. de Clerck, editors, Finite Geometries and Combinatorics, page 277",
          "text": "We may ask in general which Coxeter groups arise as subgroups of a given Coxeter group. This question is of course far too general. However, there are Coxeter groups which arise canonically as subgroups of a given Coxeter group.",
          "type": "quotation"
        },
        {
          "ref": "2007, Michael B. Smythe, Julian Webster, “12: Discrete Spatial Models”, in Marco Aiello, Ian Pratt-Hartmann, Johan van Benthem, editors, Handbook of Spatial Logics, page 795",
          "text": "The above definitions, of Cayley graph, reflection, wall, half-space, folding, Bruhat order, etc., work for any Coxeter group.",
          "type": "quotation"
        },
        {
          "ref": "2012, Daniel Allcock, “The Reflective Lorentzian Lattices of Rank 3”, in Memoirs of the American Mathematical Society, number 1033, page vii",
          "text": "The problem of classifying all reflective lattices of given rank is also of interest in its own right from the perspectives of Coxeter groups and arithmetic subgroups of O(n,1).",
          "type": "quotation"
        }
      ],
      "glosses": [
        "Any of a class of groups whose finite cases are precisely the finite reflection groups (including the symmetry groups of polytopes), but which are more varied in their infinite cases, and whose range of application encompasses various areas of mathematics."
      ],
      "links": [
        [
          "mathematics",
          "mathematics"
        ],
        [
          "geometry",
          "geometry"
        ],
        [
          "group theory",
          "group theory"
        ],
        [
          "class",
          "class"
        ],
        [
          "group",
          "group"
        ],
        [
          "finite",
          "finite"
        ],
        [
          "reflection group",
          "reflection group"
        ],
        [
          "symmetry group",
          "symmetry group"
        ],
        [
          "polytope",
          "polytope"
        ],
        [
          "application",
          "application"
        ]
      ],
      "raw_glosses": [
        "(mathematics, geometry, group theory) Any of a class of groups whose finite cases are precisely the finite reflection groups (including the symmetry groups of polytopes), but which are more varied in their infinite cases, and whose range of application encompasses various areas of mathematics."
      ],
      "topics": [
        "geometry",
        "group-theory",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "Coxeter group",
        "Harold Scott MacDonald Coxeter"
      ]
    }
  ],
  "word": "Coxeter group"
}