{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "DA515YH1J0"
    },
    {
      "field": "instance_of",
      "target": "M0WAT1VQZH"
    }
  ],
  "body": "\n## Summary\n\nA [[group|G5S1M9]] is residually finite if every element other than the identity survives in some finite quotient.\nSo the group can be studied through its finite images, which separate all its elements.\n[[Free groups|0KTW26C80J]], finitely generated abelian groups and finitely generated subgroups of $GL(n, \\C)$ are residually finite.\nA finitely generated residually finite group is [[Hopfian|PHEN4E1JPT]], and a finitely presented residually finite group has solvable [[word problem|PBGTTAVMMT]].\n\n## Explanations\n\n### L1 human\n\nTo see that an element $g$ is not the identity, map the group onto a finite group and look at the image of $g$.\nIn a residually finite group this always works for a suitable finite group.\nFor example, the integer $n \\ne 0$ survives in $\\Z / (\\abs{n} + 1)$, since $0 \u003c \\abs{n} \u003c \\abs{n} + 1$.\nThe group of rational numbers fails badly: it has no finite quotient except the trivial one.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "kind": "concept",
    "msc": [
      "20E26"
    ],
    "name": "Residually finite group",
    "same_as": {
      "wikidata": "Q3777935",
      "wikipedia": "Residually finite group"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Residually_finite_group"
      }
    ],
    "specializes": [
      "G5S1M9"
    ],
    "tag": "HEE922EHQ8"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/HEE922EHQ8/",
  "links": [
    {
      "field": "specializes",
      "target": "G5S1M9"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/HEE922EHQ8.md",
  "msc": [
    "20E26"
  ],
  "name": "Residually finite group",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/HEE/HEE922EHQ8.md",
  "status": null,
  "tag": "HEE922EHQ8",
  "uri": "m2://HEE922EHQ8",
  "url": "https://math2.tamnd.com/wiki/residually-finite-group/",
  "writing": {
    "tag": "HEE922EHQ8",
    "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
      }
    ]
  }
}