"permutation group" meaning in English

{
  "forms": [
    {
      "form": "permutation groups",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "permutation group (plural permutation 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": "topical",
          "langcode": "en",
          "name": "Algebra",
          "orig": "en:Algebra",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
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          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Group theory",
          "orig": "en:Group theory",
          "parents": [
            "Algebra",
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
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          "source": "w"
        }
      ],
      "examples": [
        {
          "ref": "1979, Norman L. Biggs, A. T. White, Permutation Groups and Combinatorial Structures, page 80",
          "text": "In this chapter we shall be concerned with the relationship between permutation groups and graphs. We begin by explaining how a transitive permutation group may be represented graphically, and then we reverse the process, showing that a graph gives rise to a permutation group.",
          "type": "quotation"
        },
        {
          "ref": "1996, Helmut Volklein, Groups as Galois Groups: An Introduction, page 47",
          "text": "The Galois group G(L_f /C(x)) is called the monodromy group of f, denoted Mon(f), and viewed as a permutation group on the conjugates of y over C(x).",
          "type": "quotation"
        },
        {
          "ref": "2002, Peter J. Cameron, “B.5 Permutation Groups”, in Alexander V. Mikhalev, Günter F. Pilz, editors, The Concise Handbook of Algebra, page 86",
          "text": "Now, groups are axiomatically defined, and the above concept is a permutation group, that is, a subgroup of the symmetric group.[…]The study of finite permutation groups is one of the oldest parts of group theory, motivated initially by its connection with solvability of equations.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A group whose elements are permutations (self-bijections) of a given set and whose group operation is function composition."
      ],
      "hyponyms": [
        {
          "sense": "group whose elements are permutations of a set",
          "word": "alternating group"
        },
        {
          "sense": "group whose elements are permutations of a set",
          "word": "symmetric group"
        }
      ],
      "id": "en-permutation_group-en-noun-eP6VJeSD",
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "group theory",
          "group theory"
        ],
        [
          "group",
          "group"
        ],
        [
          "element",
          "element"
        ],
        [
          "permutation",
          "permutation"
        ],
        [
          "bijection",
          "bijection"
        ],
        [
          "set",
          "set"
        ],
        [
          "operation",
          "operation"
        ],
        [
          "composition",
          "composition"
        ]
      ],
      "raw_glosses": [
        "(algebra, group theory) A group whose elements are permutations (self-bijections) of a given set and whose group operation is function composition."
      ],
      "related": [
        {
          "word": "permutation representation"
        }
      ],
      "synonyms": [
        {
          "sense": "group whose elements are permutations of a set",
          "word": "transformation group"
        }
      ],
      "topics": [
        "algebra",
        "group-theory",
        "mathematics",
        "sciences"
      ],
      "translations": [
        {
          "code": "fi",
          "lang": "Finnish",
          "sense": "group whose elements are permutations",
          "word": "permutaatioryhmä"
        },
        {
          "code": "fr",
          "lang": "French",
          "sense": "group whose elements are permutations",
          "tags": [
            "masculine"
          ],
          "word": "groupe de permutations"
        }
      ],
      "wikipedia": [
        "permutation group"
      ]
    }
  ],
  "word": "permutation group"
}

{
  "forms": [
    {
      "form": "permutation groups",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "permutation group (plural permutation groups)",
      "name": "en-noun"
    }
  ],
  "hyponyms": [
    {
      "sense": "group whose elements are permutations of a set",
      "word": "alternating group"
    },
    {
      "sense": "group whose elements are permutations of a set",
      "word": "symmetric group"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "permutation representation"
    }
  ],
  "senses": [
    {
      "categories": [
        "English countable nouns",
        "English entries with incorrect language header",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms with quotations",
        "en:Algebra",
        "en:Group theory"
      ],
      "examples": [
        {
          "ref": "1979, Norman L. Biggs, A. T. White, Permutation Groups and Combinatorial Structures, page 80",
          "text": "In this chapter we shall be concerned with the relationship between permutation groups and graphs. We begin by explaining how a transitive permutation group may be represented graphically, and then we reverse the process, showing that a graph gives rise to a permutation group.",
          "type": "quotation"
        },
        {
          "ref": "1996, Helmut Volklein, Groups as Galois Groups: An Introduction, page 47",
          "text": "The Galois group G(L_f /C(x)) is called the monodromy group of f, denoted Mon(f), and viewed as a permutation group on the conjugates of y over C(x).",
          "type": "quotation"
        },
        {
          "ref": "2002, Peter J. Cameron, “B.5 Permutation Groups”, in Alexander V. Mikhalev, Günter F. Pilz, editors, The Concise Handbook of Algebra, page 86",
          "text": "Now, groups are axiomatically defined, and the above concept is a permutation group, that is, a subgroup of the symmetric group.[…]The study of finite permutation groups is one of the oldest parts of group theory, motivated initially by its connection with solvability of equations.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A group whose elements are permutations (self-bijections) of a given set and whose group operation is function composition."
      ],
      "links": [
        [
          "algebra",
          "algebra"
        ],
        [
          "group theory",
          "group theory"
        ],
        [
          "group",
          "group"
        ],
        [
          "element",
          "element"
        ],
        [
          "permutation",
          "permutation"
        ],
        [
          "bijection",
          "bijection"
        ],
        [
          "set",
          "set"
        ],
        [
          "operation",
          "operation"
        ],
        [
          "composition",
          "composition"
        ]
      ],
      "raw_glosses": [
        "(algebra, group theory) A group whose elements are permutations (self-bijections) of a given set and whose group operation is function composition."
      ],
      "topics": [
        "algebra",
        "group-theory",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "permutation group"
      ]
    }
  ],
  "synonyms": [
    {
      "sense": "group whose elements are permutations of a set",
      "word": "transformation group"
    }
  ],
  "translations": [
    {
      "code": "fi",
      "lang": "Finnish",
      "sense": "group whose elements are permutations",
      "word": "permutaatioryhmä"
    },
    {
      "code": "fr",
      "lang": "French",
      "sense": "group whose elements are permutations",
      "tags": [
        "masculine"
      ],
      "word": "groupe de permutations"
    }
  ],
  "word": "permutation group"
}