{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "EESNTXFWHZ"
    },
    {
      "field": "uses",
      "target": "V7ERB5XHQR"
    },
    {
      "field": "defines",
      "target": "VABPHNZAVV"
    }
  ],
  "body": "\n## Summary\n\nA profinite group is a topological group that is an inverse limit of finite groups.\nEquivalently, it is a compact, Hausdorff and totally disconnected topological group.\nProfinite groups record a whole compatible family of finite quotients at once.\nThe main examples are the [[p-adic integers|V6TJAFZ4MD]], the profinite completion $\\hat\\Z$ of $\\Z$, and Galois groups of infinite Galois extensions with the Krull topology.\n\n## Explanations\n\n### L1 human\n\nThe integers modulo $p$, modulo $p^2$, modulo $p^3$, and so on, form a tower of finite groups, each mapping onto the previous one by reduction.\nA compatible choice of one residue at each level is a $p$-adic integer, and these choices form a group.\nThis group is profinite: it is built entirely from finite pieces, and it carries a topology in which two elements are close when they agree at many levels.\nIn the same way, the Galois group of all algebraic numbers over $\\Q$ is assembled from the finite Galois groups of number fields.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "20E18",
      "22C05"
    ],
    "name": "Profinite group",
    "same_as": {
      "wikidata": "Q1428923",
      "wikipedia": "Profinite group"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Profinite_group"
      }
    ],
    "tag": "KA0YDB6QXE"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/KA0YDB6QXE/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/KA0YDB6QXE.md",
  "msc": [
    "20E18",
    "22C05"
  ],
  "name": "Profinite group",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/KA0/KA0YDB6QXE.md",
  "status": null,
  "tag": "KA0YDB6QXE",
  "uri": "m2://KA0YDB6QXE",
  "url": "https://math2.tamnd.com/wiki/profinite-group/",
  "writing": {
    "tag": "KA0YDB6QXE",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}