"quotient group" meaning in English

{
  "forms": [
    {
      "form": "quotient groups",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "quotient group (plural quotient 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": "Group theory",
          "orig": "en:Group theory",
          "parents": [
            "Algebra",
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "examples": [
        {
          "ref": "1975, John R. Stallings, “Quotients of the Powers of the Augmentation Ideal in a Group Ring”, in Lee Paul Neuwirth, editor, Knots, Groups, and 3-manifolds: Papers Dedicated to the Memory of R. H. Fox, Princeton University Press, page 101",
          "text": "This paper shows how to compute the quotient groups Jⁿ/Jⁿ⁺¹ (as well as the multiplicative structure of the graded ring consisting of these quotient groups).",
          "type": "quotation"
        },
        {
          "ref": "1983, David H. Sattinger, Branching in the Presence of Symmetry, Society for Industrial and Applied Mathematics, page 33",
          "text": "The Weyl group is the quotient group N_H/T_H, and in the present case the Weyl group is simply the permutation group S₃.",
          "type": "quotation"
        },
        {
          "ref": "2002, Alexander Arhangel'skii, “Topological Invariants in Algebraic Environment”, in Miroslav Hušek, Jan van Mill, editors, Recent Progress in General Topology II, North-Holland: Elsevier, page 39",
          "text": "The class of reflexive groups doesn't behave nicely with regards to operations: a closed subgroup of a reflexive group need not be reflexive, and a quotient group of a reflexive group need not be reflexive.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A group obtained from a larger group by aggregating elements via an equivalence relation that preserves group structure."
      ],
      "id": "en-quotient_group-en-noun-RvTIu~1h",
      "links": [
        [
          "group theory",
          "group theory"
        ],
        [
          "group",
          "group"
        ],
        [
          "equivalence relation",
          "equivalence relation"
        ]
      ],
      "raw_glosses": [
        "(group theory) A group obtained from a larger group by aggregating elements via an equivalence relation that preserves group structure."
      ],
      "synonyms": [
        {
          "sense": "group obtained from a larger group by aggregating elements",
          "word": "factor group"
        }
      ],
      "topics": [
        "group-theory",
        "mathematics",
        "sciences"
      ],
      "translations": [
        {
          "code": "fi",
          "lang": "Finnish",
          "sense": "group obtained from a larger group by aggregating elements",
          "word": "tekijäryhmä"
        },
        {
          "code": "it",
          "lang": "Italian",
          "sense": "group obtained from a larger group by aggregating elements",
          "tags": [
            "masculine"
          ],
          "word": "gruppo quoziente"
        }
      ],
      "wikipedia": [
        "quotient group"
      ]
    }
  ],
  "word": "quotient group"
}

{
  "forms": [
    {
      "form": "quotient groups",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "quotient group (plural quotient groups)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        "English countable nouns",
        "English entries with incorrect language header",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms with quotations",
        "en:Group theory"
      ],
      "examples": [
        {
          "ref": "1975, John R. Stallings, “Quotients of the Powers of the Augmentation Ideal in a Group Ring”, in Lee Paul Neuwirth, editor, Knots, Groups, and 3-manifolds: Papers Dedicated to the Memory of R. H. Fox, Princeton University Press, page 101",
          "text": "This paper shows how to compute the quotient groups Jⁿ/Jⁿ⁺¹ (as well as the multiplicative structure of the graded ring consisting of these quotient groups).",
          "type": "quotation"
        },
        {
          "ref": "1983, David H. Sattinger, Branching in the Presence of Symmetry, Society for Industrial and Applied Mathematics, page 33",
          "text": "The Weyl group is the quotient group N_H/T_H, and in the present case the Weyl group is simply the permutation group S₃.",
          "type": "quotation"
        },
        {
          "ref": "2002, Alexander Arhangel'skii, “Topological Invariants in Algebraic Environment”, in Miroslav Hušek, Jan van Mill, editors, Recent Progress in General Topology II, North-Holland: Elsevier, page 39",
          "text": "The class of reflexive groups doesn't behave nicely with regards to operations: a closed subgroup of a reflexive group need not be reflexive, and a quotient group of a reflexive group need not be reflexive.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A group obtained from a larger group by aggregating elements via an equivalence relation that preserves group structure."
      ],
      "links": [
        [
          "group theory",
          "group theory"
        ],
        [
          "group",
          "group"
        ],
        [
          "equivalence relation",
          "equivalence relation"
        ]
      ],
      "raw_glosses": [
        "(group theory) A group obtained from a larger group by aggregating elements via an equivalence relation that preserves group structure."
      ],
      "topics": [
        "group-theory",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "quotient group"
      ]
    }
  ],
  "synonyms": [
    {
      "sense": "group obtained from a larger group by aggregating elements",
      "word": "factor group"
    }
  ],
  "translations": [
    {
      "code": "fi",
      "lang": "Finnish",
      "sense": "group obtained from a larger group by aggregating elements",
      "word": "tekijäryhmä"
    },
    {
      "code": "it",
      "lang": "Italian",
      "sense": "group obtained from a larger group by aggregating elements",
      "tags": [
        "masculine"
      ],
      "word": "gruppo quoziente"
    }
  ],
  "word": "quotient group"
}