"free group" meaning in English

{
  "forms": [
    {
      "form": "free groups",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "free group (plural free 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": [
        {
          "text": "Given a set S of \"free generators\" of a free group, let S⁻¹ be the set of inverses of the generators, which are in one-to-one correspondence with the generators (the two sets are disjoint), then let (S∪S⁻¹)^* be the Kleene closure of the union of those two sets. For any string w in the Kleene closure let r(w) be its reduced form, obtained by cutting out any occurrences of the form xx⁻¹ or x⁻¹x where x∈S. Noting that r(r(w)) = r(w) for any string w, define an equivalence relation ∼ such that u∼v if and only if r(u)=r(v). Then let the underlying set of the free group generated by S be the quotient set (S∪S⁻¹)^*/∼ and let its operator be concatenation followed by reduction."
        },
        {
          "ref": "1999, John R. Stallings, “Whitehead graphs on handlebodies”, in John Cossey, Charles F. Miller, Michael Shapiro, Walter D. Neumann, editors, Geometric Group Theory Down Under: Proceedings of a Special Year in Geometric Group Theory, Walter de Gruyter, page 317",
          "text": "A subset A of a free group F is called \"separable\" when there is a non-trivial free factorization F = F₁ * F₂ such that each element of A is conjugate to an element of F₁ or of F₂.",
          "type": "quotation"
        },
        {
          "text": "2002, Gilbert Baumslag, B.9 Free and Relatively Free Groups, Alexander V. Mikhalev, Günter F. Pilz, The Concise Handbook of Algebra, Kluwer Academic, page 102,\nThe free groups in V then all take the form H/V(H), where H is a suitably chosen absolutely free group."
        },
        {
          "ref": "2006, Anthony W. Knapp, Basic Algebra, Springer, page 303",
          "text": "The context for generators and relations is that of a free group on the set of generators, and the relations indicate passage to a quotient of this free group by a normal subgroup.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A group that has a presentation without relators; equivalently, a free product of some number of copies of ℤ."
      ],
      "id": "en-free_group-en-noun-K-gCGrgC",
      "links": [
        [
          "group theory",
          "group theory"
        ],
        [
          "group",
          "group"
        ],
        [
          "presentation",
          "presentation"
        ],
        [
          "relator",
          "relator"
        ],
        [
          "free product",
          "free product"
        ],
        [
          "copies",
          "copy"
        ],
        [
          "ℤ",
          "ℤ"
        ]
      ],
      "raw_glosses": [
        "(group theory) A group that has a presentation without relators; equivalently, a free product of some number of copies of ℤ."
      ],
      "topics": [
        "group-theory",
        "mathematics",
        "sciences"
      ],
      "translations": [
        {
          "code": "is",
          "lang": "Icelandic",
          "sense": "group whose presentation consists of generators",
          "tags": [
            "feminine"
          ],
          "word": "frjáls grúpa"
        },
        {
          "code": "it",
          "lang": "Italian",
          "sense": "group whose presentation consists of generators",
          "tags": [
            "masculine"
          ],
          "word": "gruppo libero"
        }
      ],
      "wikipedia": [
        "free group"
      ]
    }
  ],
  "word": "free group"
}

{
  "forms": [
    {
      "form": "free groups",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "free group (plural free 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": [
        {
          "text": "Given a set S of \"free generators\" of a free group, let S⁻¹ be the set of inverses of the generators, which are in one-to-one correspondence with the generators (the two sets are disjoint), then let (S∪S⁻¹)^* be the Kleene closure of the union of those two sets. For any string w in the Kleene closure let r(w) be its reduced form, obtained by cutting out any occurrences of the form xx⁻¹ or x⁻¹x where x∈S. Noting that r(r(w)) = r(w) for any string w, define an equivalence relation ∼ such that u∼v if and only if r(u)=r(v). Then let the underlying set of the free group generated by S be the quotient set (S∪S⁻¹)^*/∼ and let its operator be concatenation followed by reduction."
        },
        {
          "ref": "1999, John R. Stallings, “Whitehead graphs on handlebodies”, in John Cossey, Charles F. Miller, Michael Shapiro, Walter D. Neumann, editors, Geometric Group Theory Down Under: Proceedings of a Special Year in Geometric Group Theory, Walter de Gruyter, page 317",
          "text": "A subset A of a free group F is called \"separable\" when there is a non-trivial free factorization F = F₁ * F₂ such that each element of A is conjugate to an element of F₁ or of F₂.",
          "type": "quotation"
        },
        {
          "text": "2002, Gilbert Baumslag, B.9 Free and Relatively Free Groups, Alexander V. Mikhalev, Günter F. Pilz, The Concise Handbook of Algebra, Kluwer Academic, page 102,\nThe free groups in V then all take the form H/V(H), where H is a suitably chosen absolutely free group."
        },
        {
          "ref": "2006, Anthony W. Knapp, Basic Algebra, Springer, page 303",
          "text": "The context for generators and relations is that of a free group on the set of generators, and the relations indicate passage to a quotient of this free group by a normal subgroup.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A group that has a presentation without relators; equivalently, a free product of some number of copies of ℤ."
      ],
      "links": [
        [
          "group theory",
          "group theory"
        ],
        [
          "group",
          "group"
        ],
        [
          "presentation",
          "presentation"
        ],
        [
          "relator",
          "relator"
        ],
        [
          "free product",
          "free product"
        ],
        [
          "copies",
          "copy"
        ],
        [
          "ℤ",
          "ℤ"
        ]
      ],
      "raw_glosses": [
        "(group theory) A group that has a presentation without relators; equivalently, a free product of some number of copies of ℤ."
      ],
      "topics": [
        "group-theory",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "free group"
      ]
    }
  ],
  "translations": [
    {
      "code": "is",
      "lang": "Icelandic",
      "sense": "group whose presentation consists of generators",
      "tags": [
        "feminine"
      ],
      "word": "frjáls grúpa"
    },
    {
      "code": "it",
      "lang": "Italian",
      "sense": "group whose presentation consists of generators",
      "tags": [
        "masculine"
      ],
      "word": "gruppo libero"
    }
  ],
  "word": "free group"
}