"permutation group" meaning in All languages combined

See permutation group on Wiktionary

Noun [English]

Forms: permutation groups [plural]
Head templates: {{en-noun}} permutation group (plural permutation groups)
  1. (algebra, group theory) A group whose elements are permutations (self-bijections) of a given set and whose group operation is function composition. Wikipedia link: permutation group Categories (topical): Algebra, Group theory Synonyms (group whose elements are permutations of a set): transformation group Hyponyms (group whose elements are permutations of a set): alternating group, symmetric group Related terms: permutation representation Translations (group whose elements are permutations): permutaatioryhmä (Finnish), groupe de permutations [masculine] (French)

Inflected forms

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      "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": "quote"
        },
        {
          "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": "quote"
        },
        {
          "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": "quote"
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        "A group whose elements are permutations (self-bijections) of a given set and whose group operation is function composition."
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          "sense": "group whose elements are permutations of a set",
          "word": "alternating group"
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        "(algebra, group theory) A group whose elements are permutations (self-bijections) of a given set and whose group operation is function composition."
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        {
          "code": "fi",
          "lang": "Finnish",
          "sense": "group whose elements are permutations",
          "word": "permutaatioryhmä"
        },
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          "code": "fr",
          "lang": "French",
          "sense": "group whose elements are permutations",
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          "word": "groupe de permutations"
        }
      ],
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  "word": "permutation group"
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          "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": "quote"
        },
        {
          "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": "quote"
        },
        {
          "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.",
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        "(algebra, group theory) A group whose elements are permutations (self-bijections) of a given set and whose group operation is function composition."
      ],
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        "group-theory",
        "mathematics",
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    {
      "sense": "group whose elements are permutations of a set",
      "word": "transformation group"
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  "translations": [
    {
      "code": "fi",
      "lang": "Finnish",
      "sense": "group whose elements are permutations",
      "word": "permutaatioryhmä"
    },
    {
      "code": "fr",
      "lang": "French",
      "sense": "group whose elements are permutations",
      "tags": [
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      ],
      "word": "groupe de permutations"
    }
  ],
  "word": "permutation group"
}

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This page is a part of the kaikki.org machine-readable All languages combined dictionary. This dictionary is based on structured data extracted on 2025-01-03 from the enwiktionary dump dated 2025-01-01 using wiktextract (eaedd02 and 8fbd9e8). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.

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