"quotient group" meaning in English

{
  "forms": [
    {
      "form": "quotient groups",
      "tags": [
        "plural"
      ]
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      "expansion": "quotient group (plural quotient groups)",
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  "lang_code": "en",
  "pos": "noun",
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          "langcode": "en",
          "name": "Group theory",
          "orig": "en:Group theory",
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      "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": "quote"
        },
        {
          "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": "quote"
        },
        {
          "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": "quote"
        }
      ],
      "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",
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          "group theory",
          "group theory"
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          "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"
}

{
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    {
      "form": "quotient groups",
      "tags": [
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
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        {
          "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": "quote"
        },
        {
          "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": "quote"
        },
        {
          "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": "quote"
        }
      ],
      "glosses": [
        "A group obtained from a larger group by aggregating elements via an equivalence relation that preserves group structure."
      ],
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        [
          "group theory",
          "group theory"
        ],
        [
          "group",
          "group"
        ],
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          "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"
}