{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $G$ be a [[topological group|92700QAB5Y]] with identity element $e$.\nThe *identity component* of $G$, written $G_0$ or $G^\\circ$, is the [[connected component|0JG9W3KA6B]] of $G$ that contains $e$.\n\n## Remarks\n\n$G_0$ is a closed [[normal subgroup|0CB0DX5BVV]] of $G$.\nIt is a subgroup because $G_0 G_0^{-1}$ is the image of the connected set $G_0 \\times G_0$ under $(x, y) \\mapsto xy^{-1}$ and contains $e$, so it lies in $G_0$.\nIt is normal because each inner automorphism $x \\mapsto gxg^{-1}$ is a homeomorphism fixing $e$.\nThe [[quotient|417ZRDCHY6]] $G/G_0$ is a [[totally disconnected group|AW3J9AQSA0]].\nExample: for $G = \\mathrm{GL}_n(\\R)$, the identity component is the subgroup of matrices with positive determinant.\nFor $G = (\\Q, +)$ with the topology from $\\R$, the identity component is $\\set{0}$, since $\\Q$ is totally disconnected.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "SDQTNBDJ2H",
    "kind": "definition",
    "msc": [
      "22A05",
      "54D05"
    ],
    "name": "Identity component",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Identity_component"
      }
    ],
    "tag": "QMYZ798EAR",
    "uses": [
      "92700QAB5Y",
      "0JG9W3KA6B",
      "0CB0DX5BVV",
      "417ZRDCHY6",
      "AW3J9AQSA0"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/QMYZ798EAR/",
  "links": [
    {
      "field": "defines",
      "target": "SDQTNBDJ2H"
    },
    {
      "field": "uses",
      "target": "92700QAB5Y"
    },
    {
      "field": "uses",
      "target": "0JG9W3KA6B"
    },
    {
      "field": "uses",
      "target": "0CB0DX5BVV"
    },
    {
      "field": "uses",
      "target": "417ZRDCHY6"
    },
    {
      "field": "uses",
      "target": "AW3J9AQSA0"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/QMYZ798EAR.md",
  "msc": [
    "22A05",
    "54D05"
  ],
  "name": "Identity component",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/QMY/QMYZ798EAR.md",
  "status": null,
  "tag": "QMYZ798EAR",
  "uri": "m2://QMYZ798EAR",
  "url": "https://tamnd.github.io/math2/wiki/identity-component-definition/",
  "writing": {
    "tag": "QMYZ798EAR",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}