{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet a [[group|G5S1M9]] $G$ [[act|0WCY42EV9V]] on a set $X$, let $x \\in X$, and let $g \\in G$.\nThen the [[stabiliser|0AYMP41THN]] of $g \\cdot x$ is $G_{g \\cdot x} = g G_x g^{-1}$.\n\n## Remarks\n\nAn element $h$ fixes $g \\cdot x$ if and only if $g^{-1}hg$ fixes $x$.\nSo the stabilisers of the points of one [[orbit|0YQPZB5Q1G]] are [[conjugate subgroups|0H11ZPX4ND]].\nIn particular, they all have the same order.\n\n## Explanations\n\n### L1 human\n\nIf $g$ moves $x$ to $y$, then the elements that fix $y$ are the elements that fix $x$, seen through $g$.\nTo fix $y$, you can move $y$ back to $x$, fix $x$, and move $x$ to $y$ again.\nSo the two stabilisers are conjugate.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "kind": "statement",
    "msc": [
      "20B05"
    ],
    "name": "Stabilisers of points in the same orbit are conjugate",
    "role": "proposition",
    "sources": [
      {
        "locator": "Fixed points and stabilizer subgroups",
        "url": "https://en.wikipedia.org/wiki/Group_action"
      }
    ],
    "tag": "05BM304NRX",
    "uses": [
      "0RGGF8ZS6D",
      "00F2JFA41J",
      "0D6WHTTH7T"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/05BM304NRX/",
  "links": [
    {
      "field": "uses",
      "target": "0RGGF8ZS6D"
    },
    {
      "field": "uses",
      "target": "00F2JFA41J"
    },
    {
      "field": "uses",
      "target": "0D6WHTTH7T"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/05BM304NRX.md",
  "msc": [
    "20B05"
  ],
  "name": "Stabilisers of points in the same orbit are conjugate",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/05B/05BM304NRX.md",
  "status": null,
  "tag": "05BM304NRX",
  "uri": "m2://05BM304NRX",
  "url": "https://math2.tamnd.com/wiki/stabilisers-of-points-in-the-same-orbit-are-conjugate/",
  "writing": {
    "tag": "05BM304NRX",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}