{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet a [[group|G5S1M9]] $G$ [[act|0WCY42EV9V]] on a finite set $X$, and let $x \\in X$.\nWrite $G \\cdot x$ for the [[orbit|0YQPZB5Q1G]] of $x$, and $G_y$ for the [[stabiliser|0AYMP41THN]] of a point $y$.\nThen the sizes of the stabilisers of all points of the orbit add up to the order of the group:\n\n$$\\sum_{y \\in G \\cdot x} |G_y| = |G|.$$\n\n## Remarks\n\nThe stabilisers of the points of one orbit are conjugate, so they all have the same size.\nBy the [[orbit-stabiliser theorem|0E2HSCKZ6C]], this common size is $|G| / |G \\cdot x|$.\nThe orbit has $|G \\cdot x|$ terms, so the sum is $|G|$.\n\nFor example, let $G = \\mathbb{Z}/4\\mathbb{Z}$ rotate the corners of a square. The orbit of a corner has $4$ points, each with trivial stabiliser, so the sum is $4 = |G|$.\n",
  "commit": "bdea0329dc328964ec62566e23a0159c6ce93f2d",
  "front": {
    "kind": "statement",
    "msc": [
      "20B05",
      "05E18"
    ],
    "name": "Stabiliser sizes over an orbit add up to the order of the group",
    "role": "proposition",
    "sources": [
      {
        "locator": "source/s_polya_burnside.ptx:9",
        "url": "https://github.com/mitchkeller/applied-combinatorics/blob/ebd95d6bd59e23c633c65cf560bcb93efa912aae/source/s_polya_burnside.ptx#L9"
      }
    ],
    "tag": "CQTDS8B6R7",
    "uses": [
      "G5S1M9",
      "0WCY42EV9V",
      "0YQPZB5Q1G",
      "0AYMP41THN",
      "0E2HSCKZ6C"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/CQTDS8B6R7/",
  "links": [
    {
      "field": "uses",
      "target": "G5S1M9"
    },
    {
      "field": "uses",
      "target": "0WCY42EV9V"
    },
    {
      "field": "uses",
      "target": "0YQPZB5Q1G"
    },
    {
      "field": "uses",
      "target": "0AYMP41THN"
    },
    {
      "field": "uses",
      "target": "0E2HSCKZ6C"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/CQTDS8B6R7.md",
  "msc": [
    "20B05",
    "05E18"
  ],
  "name": "Stabiliser sizes over an orbit add up to the order of the group",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/CQT/CQTDS8B6R7.md",
  "status": null,
  "tag": "CQTDS8B6R7",
  "uri": "m2://CQTDS8B6R7",
  "url": "https://math2.tamnd.com/wiki/stabiliser-sizes-over-an-orbit-add-up-to-the-order-of-the-group/",
  "writing": {
    "tag": "CQTDS8B6R7",
    "kind": "statement",
    "level": "W1",
    "next": [
      {
        "what": "edge about out",
        "note": "the concepts that the statement is about",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge proves in",
        "note": "at least one proof node",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge instance_of or illustrates in",
        "note": "one example that shows the statement at work",
        "have": 0,
        "want": 1
      },
      {
        "what": "explanation L2",
        "have": 0,
        "want": 1
      }
    ]
  }
}