"profinite group" meaning in English

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  "lang_code": "en",
  "pos": "noun",
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      "examples": [
        {
          "ref": "2000, Haruzo Hida, Modular Forms and Galois Cohomology, Cambridge University Press, page 28:",
          "text": "This proposition shows that to have a representation with big image of a profinite group, for example, a Galois group, we need to take a profinite topological ring A as a coefficient ring, like the p-adic integer ring #x5C;Z#x5F;p.",
          "type": "quote"
        },
        {
          "ref": "2011, Moshe Jarden, Algebraic Patching, Springer, page 207:",
          "text": "We have already pointed out that a profinite group G of an infinite rank m is free of rank m if (and only if) G is projective and every finite split embedding problem for G with a nontrivial kernel has m solutions (Proposition 9.4.7). Dropping the condition on G to be projective leads to the notion of a \"quasi-free profinite group\" (Section 10.6).",
          "type": "quote"
        },
        {
          "text": "2014, David Barnes, Constanze Roitzheim, Rational equivariant rigidity, Christian Ausoni, Kathryn Hess, Brenda Johnson, Wolfgang Lück, Jérôme Scherer, An Alpine Expedition through Algebraic Topology, American Mathematical Society, page 14,\nRecall that a profinite group is an inverse limit of an inverse system of finite groups, with the p-adic numbers being the canonical example. Any finite group is, of course, profinite, but when we talk of a profinite group we assume that the group is infinite."
        }
      ],
      "glosses": [
        "A topological group that is isomorphic to the inverse limit of some inverse system of discrete finite groups; equivalently, a topological group that is also a Stone space."
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        "(topology, category theory) A topological group that is isomorphic to the inverse limit of some inverse system of discrete finite groups; equivalently, a topological group that is also a Stone space."
      ],
      "related": [
        {
          "word": "profinite"
        },
        {
          "word": "profinite integer"
        },
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      "synonyms": [
        {
          "word": "pro-finite group"
        }
      ],
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        "engineering",
        "mathematics",
        "natural-sciences",
        "physical-sciences",
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      "translations": [
        {
          "code": "fr",
          "lang": "French",
          "sense": "topological group isomorphic to the inverse limit of an inverse system of discrete finite groups",
          "tags": [
            "masculine"
          ],
          "word": "groupe profini"
        },
        {
          "code": "de",
          "lang": "German",
          "sense": "topological group isomorphic to the inverse limit of an inverse system of discrete finite groups",
          "tags": [
            "feminine"
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          "word": "proendliche Gruppe"
        }
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      "wikipedia": [
        "profinite group"
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  "word": "profinite group"
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    {
      "form": "profinite groups",
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      "word": "profinite integer"
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          "ref": "2000, Haruzo Hida, Modular Forms and Galois Cohomology, Cambridge University Press, page 28:",
          "text": "This proposition shows that to have a representation with big image of a profinite group, for example, a Galois group, we need to take a profinite topological ring A as a coefficient ring, like the p-adic integer ring #x5C;Z#x5F;p.",
          "type": "quote"
        },
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          "text": "We have already pointed out that a profinite group G of an infinite rank m is free of rank m if (and only if) G is projective and every finite split embedding problem for G with a nontrivial kernel has m solutions (Proposition 9.4.7). Dropping the condition on G to be projective leads to the notion of a \"quasi-free profinite group\" (Section 10.6).",
          "type": "quote"
        },
        {
          "text": "2014, David Barnes, Constanze Roitzheim, Rational equivariant rigidity, Christian Ausoni, Kathryn Hess, Brenda Johnson, Wolfgang Lück, Jérôme Scherer, An Alpine Expedition through Algebraic Topology, American Mathematical Society, page 14,\nRecall that a profinite group is an inverse limit of an inverse system of finite groups, with the p-adic numbers being the canonical example. Any finite group is, of course, profinite, but when we talk of a profinite group we assume that the group is infinite."
        }
      ],
      "glosses": [
        "A topological group that is isomorphic to the inverse limit of some inverse system of discrete finite groups; equivalently, a topological group that is also a Stone space."
      ],
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        "(topology, category theory) A topological group that is isomorphic to the inverse limit of some inverse system of discrete finite groups; equivalently, a topological group that is also a Stone space."
      ],
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  "synonyms": [
    {
      "word": "pro-finite group"
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    {
      "code": "fr",
      "lang": "French",
      "sense": "topological group isomorphic to the inverse limit of an inverse system of discrete finite groups",
      "tags": [
        "masculine"
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      "word": "groupe profini"
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      "sense": "topological group isomorphic to the inverse limit of an inverse system of discrete finite groups",
      "tags": [
        "feminine"
      ],
      "word": "proendliche Gruppe"
    }
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  "word": "profinite group"
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