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  "body": "\n## Statement\n\nLet $M$ be a connected [[topological manifold|QQQ9B1DK4P]] and let $G$ be a [[locally compact group|AHV05ZBCYW]].\nSuppose that $G$ acts on $M$ continuously and [[faithfully|0P7G9NGG2G]].\nThat is, the map $G \\times M \\to M$ is continuous, and only the identity of $G$ fixes every point of $M$.\nThen $G$ is a [[Lie group|7JBRGCVJRB]].\n\n## Remarks\n\nThe conjecture is a form of [[Hilbert's fifth problem|NR70GHZ37D]], which Hilbert stated for transformation groups; it is named after David Hilbert and Paul A. Smith.\nIt is known to be equivalent to the claim that, for every prime $p$, the group $\\Z_p$ of $p$-adic integers cannot act continuously and faithfully on a connected manifold.\nThe conjecture is proved for manifolds of dimension at most $3$; the case of dimension $3$ is due to John Pardon.\nIt is also proved for actions by Lipschitz maps and for actions by diffeomorphisms.\nIn general it is open.\n\n## Explanations\n\n### L1 human\n\nA group can act on a space by moving its points around, as rotations move the points of a sphere.\nThe conjecture says that if a group acts on a manifold in a continuous and faithful way, the group must be a smooth Lie group.\nIn other words, exotic groups such as the $p$-adic integers should not act continuously on a manifold unless some element other than the identity leaves every point in place.\n",
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    "about": [
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      "first_stated": "1941",
      "notes": "A strengthening of the fifth problem of David Hilbert (1900).",
      "posed_by": "Paul A. Smith"
    },
    "kind": "conjecture",
    "msc": [
      "57S05",
      "57S10",
      "22A05"
    ],
    "name": "Hilbert–Smith conjecture",
    "same_as": {
      "wikidata": "Q5761258",
      "wikipedia": "Hilbert–Smith conjecture"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hilbert%E2%80%93Smith_conjecture"
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    "tag": "DS2AS9TSZP",
    "uses": [
      "AHV05ZBCYW",
      "7JBRGCVJRB",
      "QQQ9B1DK4P",
      "0P7G9NGG2G"
    ]
  },
  "kind": "conjecture",
  "link": "https://tamnd.github.io/math2/node/DS2AS9TSZP/",
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  "markdown": "https://tamnd.github.io/math2/node/DS2AS9TSZP.md",
  "msc": [
    "57S05",
    "57S10",
    "22A05"
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  "name": "Hilbert–Smith conjecture",
  "prov": null,
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/DS2/DS2AS9TSZP.md",
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  "tag": "DS2AS9TSZP",
  "uri": "m2://DS2AS9TSZP",
  "url": "https://tamnd.github.io/math2/wiki/hilbert-smith-conjecture/",
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    "tag": "DS2AS9TSZP",
    "kind": "conjecture",
    "level": "W0",
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