"free group" meaning in All languages combined

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  "lang_code": "en",
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      "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": "quote"
        },
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          "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."
        },
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          "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.",
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        "A group that has a presentation without relators; equivalently, a free product of some number of copies of ℤ."
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        "(group theory) A group that has a presentation without relators; equivalently, a free product of some number of copies of ℤ."
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      "translations": [
        {
          "code": "is",
          "lang": "Icelandic",
          "sense": "group whose presentation consists of generators",
          "tags": [
            "feminine"
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          "word": "frjáls grúpa"
        },
        {
          "code": "it",
          "lang": "Italian",
          "sense": "group whose presentation consists of generators",
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            "masculine"
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          "word": "gruppo libero"
        }
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          "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."
        },
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          "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": "quote"
        },
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          "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."
        },
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          "ref": "2006, Anthony W. Knapp, Basic Algebra, Springer, page 303:",
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    {
      "code": "is",
      "lang": "Icelandic",
      "sense": "group whose presentation consists of generators",
      "tags": [
        "feminine"
      ],
      "word": "frjáls grúpa"
    },
    {
      "code": "it",
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      "sense": "group whose presentation consists of generators",
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        "masculine"
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      "word": "gruppo libero"
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