{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "383CHJH46R"
    }
  ],
  "body": "\n## Summary\n\nThe profinite completion of a [[group|G5S1M9]] $G$ is the inverse limit of all its finite quotient groups.\nIt is a [[profinite group|KA0YDB6QXE]] that records every finite quotient of $G$ at once, together with a natural homomorphism from $G$.\nThis homomorphism is injective exactly when $G$ is [[residually finite|HEE922EHQ8]].\nThe profinite completion of $\\Z$ is the group $\\hat\\Z$ of profinite integers, which is the product of the $p$-adic integers over all primes $p$.\n\n## Explanations\n\n### L1 human\n\nA group can be studied through its finite images, such as the integers modulo $n$ for the group of integers.\nThe profinite completion collects all of these finite pictures into one object.\nAn element of it is a compatible choice of an image in every finite quotient.\nFor the integers, it is a choice of a residue modulo every $n$, consistent under reduction, and such choices form the group $\\hat\\Z$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "20E18",
      "20E26"
    ],
    "name": "Profinite completion",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Profinite_integer"
      }
    ],
    "tag": "CG9EH0TN27"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/CG9EH0TN27/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/CG9EH0TN27.md",
  "msc": [
    "20E18",
    "20E26"
  ],
  "name": "Profinite completion",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/CG9/CG9EH0TN27.md",
  "status": null,
  "tag": "CG9EH0TN27",
  "uri": "m2://CG9EH0TN27",
  "url": "https://tamnd.github.io/math2/wiki/profinite-completion/",
  "writing": {
    "tag": "CG9EH0TN27",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}