"commutator length" meaning in All languages combined

{
  "forms": [
    {
      "form": "commutator lengths",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "commutator length (plural commutator lengths)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        {
          "_dis": "53 47",
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [],
          "source": "w+disamb"
        },
        {
          "_dis": "52 48",
          "kind": "other",
          "name": "Pages with 1 entry",
          "parents": [],
          "source": "w+disamb"
        },
        {
          "_dis": "51 49",
          "kind": "other",
          "name": "Pages with entries",
          "parents": [],
          "source": "w+disamb"
        }
      ],
      "examples": [
        {
          "bold_text_offsets": [
            [
              44,
              61
            ]
          ],
          "text": "In #92;langlea,b#124;b²#61;1#92;rangle, the commutator length of aba#123;-2#125;ba is two, since it equals (aba#123;-1#125;b#123;-1#125;)(b#123;-1#125;a#123;-1#125;ba) but isn't a commutator.",
          "type": "example"
        }
      ],
      "glosses": [
        "The number of multiplicands needed, at a minimum, to express a given group element as a product of commutators."
      ],
      "id": "en-commutator_length-en-noun-7hLoCTUl",
      "links": [
        [
          "multiplicand",
          "multiplicand"
        ],
        [
          "group",
          "group"
        ],
        [
          "element",
          "element"
        ]
      ]
    },
    {
      "categories": [
        {
          "_dis": "53 47",
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [],
          "source": "w+disamb"
        },
        {
          "_dis": "52 48",
          "kind": "other",
          "name": "Pages with 1 entry",
          "parents": [],
          "source": "w+disamb"
        },
        {
          "_dis": "51 49",
          "kind": "other",
          "name": "Pages with entries",
          "parents": [],
          "source": "w+disamb"
        }
      ],
      "derived": [
        {
          "_dis1": "29 71",
          "word": "stable commutator length"
        }
      ],
      "examples": [
        {
          "bold_text_offsets": [
            [
              4,
              21
            ],
            [
              106,
              123
            ]
          ],
          "text": "The commutator length of #92;langlea,b#124;b²#61;1#92;rangle is at least two, since there's an element of commutator length two in it.",
          "type": "example"
        }
      ],
      "glosses": [
        "The supremum, over all elements of a given group's derived subgroup, of their commutator lengths."
      ],
      "id": "en-commutator_length-en-noun-NTEc3MPm",
      "links": [
        [
          "supremum",
          "supremum"
        ],
        [
          "derived subgroup",
          "derived subgroup"
        ]
      ]
    }
  ],
  "word": "commutator length"
}

{
  "categories": [
    "English countable nouns",
    "English entries with incorrect language header",
    "English lemmas",
    "English multiword terms",
    "English nouns",
    "Pages with 1 entry",
    "Pages with entries"
  ],
  "derived": [
    {
      "word": "stable commutator length"
    }
  ],
  "forms": [
    {
      "form": "commutator lengths",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "commutator length (plural commutator lengths)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        "English terms with usage examples"
      ],
      "examples": [
        {
          "bold_text_offsets": [
            [
              44,
              61
            ]
          ],
          "text": "In #92;langlea,b#124;b²#61;1#92;rangle, the commutator length of aba#123;-2#125;ba is two, since it equals (aba#123;-1#125;b#123;-1#125;)(b#123;-1#125;a#123;-1#125;ba) but isn't a commutator.",
          "type": "example"
        }
      ],
      "glosses": [
        "The number of multiplicands needed, at a minimum, to express a given group element as a product of commutators."
      ],
      "links": [
        [
          "multiplicand",
          "multiplicand"
        ],
        [
          "group",
          "group"
        ],
        [
          "element",
          "element"
        ]
      ]
    },
    {
      "categories": [
        "English terms with usage examples"
      ],
      "examples": [
        {
          "bold_text_offsets": [
            [
              4,
              21
            ],
            [
              106,
              123
            ]
          ],
          "text": "The commutator length of #92;langlea,b#124;b²#61;1#92;rangle is at least two, since there's an element of commutator length two in it.",
          "type": "example"
        }
      ],
      "glosses": [
        "The supremum, over all elements of a given group's derived subgroup, of their commutator lengths."
      ],
      "links": [
        [
          "supremum",
          "supremum"
        ],
        [
          "derived subgroup",
          "derived subgroup"
        ]
      ]
    }
  ],
  "word": "commutator length"
}