{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "5GNSQRHSSC"
    }
  ],
  "body": "\n## Definition\n\nLet $X$ be a set.\nAn *involution* on $X$ is a [[function|0EPAT51DXQ]] $f \\colon X \\to X$ such that $f(f(x)) = x$ for all $x \\in X$.\nEquivalently, $f \\circ f = \\id_X$, where $\\circ$ is [[composition|0THBK9V825]] and $\\id_X$ is the [[identity function|0PSKT5W817]] of $X$.\n\n## Remarks\n\nExample: complex conjugation $z \\mapsto \\bar{z}$ on $\\C$ is an involution, because $\\overline{\\overline{a + bi}} = \\overline{a - bi} = a + bi$.\n\nNon-example: the map $x \\mapsto x + 1$ on $\\Z$ is not an involution, because applying it twice gives $x + 2 \\ne x$.\n\nEvery involution is a bijection, and it is its own inverse.\nThe identity map is an involution.\nOn a finite set, the number of points not fixed by an involution is even, since they come in pairs $\\set{x, f(x)}$.\nIn a group, the map $x \\mapsto x^{-1}$ is an involution, because $(x^{-1})^{-1} = x$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "GP56NBN9RY",
    "kind": "definition",
    "msc": [
      "08A02",
      "16W10"
    ],
    "name": "Involution",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Involution_(mathematics)"
      }
    ],
    "tag": "Y0EZW47S8M",
    "uses": [
      "0EPAT51DXQ",
      "0THBK9V825",
      "0PSKT5W817"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/Y0EZW47S8M/",
  "links": [
    {
      "field": "defines",
      "target": "GP56NBN9RY"
    },
    {
      "field": "uses",
      "target": "0EPAT51DXQ"
    },
    {
      "field": "uses",
      "target": "0THBK9V825"
    },
    {
      "field": "uses",
      "target": "0PSKT5W817"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/Y0EZW47S8M.md",
  "msc": [
    "08A02",
    "16W10"
  ],
  "name": "Involution",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/Y0E/Y0EZW47S8M.md",
  "status": null,
  "tag": "Y0EZW47S8M",
  "uri": "m2://Y0EZW47S8M",
  "url": "https://tamnd.github.io/math2/wiki/involution-definition/",
  "writing": {
    "tag": "Y0EZW47S8M",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}