"cyclic group" meaning in English

{
  "forms": [
    {
      "form": "cyclic groups",
      "tags": [
        "plural"
      ]
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      "args": {},
      "expansion": "cyclic group (plural cyclic groups)",
      "name": "en-noun"
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
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          "langcode": "en",
          "name": "Group theory",
          "orig": "en:Group theory",
          "parents": [
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            "Mathematics",
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        }
      ],
      "examples": [
        {
          "text": "1986, N. S. Gopalakrishnan, University Algebra, New Age International, 2nd Edition, page 22,\nProposition 1.5.6. Any subgroup of an infinite cyclic group is also an infinite cyclic group."
        },
        {
          "ref": "2002, Serge Lang, Algebra, 3rd edition, Springer, page 24:",
          "text": "If u#x3A;G#x5F;1#x5C;rightarrow#x5C;mathbb#x7B;Z#x7D;#x2F;m#x5C;mathbb#x7B;Z#x7D; and v#x3A;G#x5F;2#x5C;rightarrow#x5C;mathbb#x7B;Z#x7D;#x2F;m#x5C;mathbb#x7B;Z#x7D; are isomorphisms of two cyclic groups with #x5C;mathbb#x7B;Z#x7D;#x2F;m#x5C;mathbb#x7B;Z#x7D;, then v#x7B;-1#x7D;#x5C;circu#x3A;G#x5F;1#x5C;rightarrowG#x5F;2 is an isomorphism.",
          "type": "quote"
        },
        {
          "text": "2003, Alexander Retakh (translator), Ėrnest Borisovich Vinberg, A Course in Algebra, [2001, Э. Б. Винберг, Курс алгебры, Factorial Press] American Mathematical Society, page 152,\nCyclic groups are the simplest groups imaginable."
        }
      ],
      "glosses": [
        "A group generated by a single element."
      ],
      "hypernyms": [
        {
          "word": "abelian group"
        }
      ],
      "id": "en-cyclic_group-en-noun-tmPwUUDY",
      "links": [
        [
          "group theory",
          "group theory"
        ],
        [
          "group",
          "group"
        ]
      ],
      "raw_glosses": [
        "(group theory) A group generated by a single element."
      ],
      "related": [
        {
          "word": "cyclic module"
        }
      ],
      "synonyms": [
        {
          "word": "monogenous group"
        }
      ],
      "topics": [
        "group-theory",
        "mathematics",
        "sciences"
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      "translations": [
        {
          "code": "it",
          "lang": "Italian",
          "sense": "group generated by a single element",
          "tags": [
            "masculine"
          ],
          "word": "gruppo ciclico"
        }
      ],
      "wikipedia": [
        "cyclic group"
      ]
    }
  ],
  "word": "cyclic group"
}

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    {
      "form": "cyclic groups",
      "tags": [
        "plural"
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      "expansion": "cyclic group (plural cyclic groups)",
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  "hypernyms": [
    {
      "word": "abelian group"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "cyclic module"
    }
  ],
  "senses": [
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        "Terms with Italian translations",
        "en:Group theory"
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      "examples": [
        {
          "text": "1986, N. S. Gopalakrishnan, University Algebra, New Age International, 2nd Edition, page 22,\nProposition 1.5.6. Any subgroup of an infinite cyclic group is also an infinite cyclic group."
        },
        {
          "ref": "2002, Serge Lang, Algebra, 3rd edition, Springer, page 24:",
          "text": "If u#x3A;G#x5F;1#x5C;rightarrow#x5C;mathbb#x7B;Z#x7D;#x2F;m#x5C;mathbb#x7B;Z#x7D; and v#x3A;G#x5F;2#x5C;rightarrow#x5C;mathbb#x7B;Z#x7D;#x2F;m#x5C;mathbb#x7B;Z#x7D; are isomorphisms of two cyclic groups with #x5C;mathbb#x7B;Z#x7D;#x2F;m#x5C;mathbb#x7B;Z#x7D;, then v#x7B;-1#x7D;#x5C;circu#x3A;G#x5F;1#x5C;rightarrowG#x5F;2 is an isomorphism.",
          "type": "quote"
        },
        {
          "text": "2003, Alexander Retakh (translator), Ėrnest Borisovich Vinberg, A Course in Algebra, [2001, Э. Б. Винберг, Курс алгебры, Factorial Press] American Mathematical Society, page 152,\nCyclic groups are the simplest groups imaginable."
        }
      ],
      "glosses": [
        "A group generated by a single element."
      ],
      "links": [
        [
          "group theory",
          "group theory"
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        [
          "group",
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        ]
      ],
      "raw_glosses": [
        "(group theory) A group generated by a single element."
      ],
      "synonyms": [
        {
          "word": "monogenous group"
        }
      ],
      "topics": [
        "group-theory",
        "mathematics",
        "sciences"
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      "wikipedia": [
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  "translations": [
    {
      "code": "it",
      "lang": "Italian",
      "sense": "group generated by a single element",
      "tags": [
        "masculine"
      ],
      "word": "gruppo ciclico"
    }
  ],
  "word": "cyclic group"
}