{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "08BPSC6YTE"
    }
  ],
  "body": "\n## Statement\n\nLet $G$ be a [[group|G5S1M9]] and let $a \\in G$.\nThen $|\\operatorname{Cl}(a)| = [G : C_G(a)]$, where $\\operatorname{Cl}(a)$ is the [[conjugacy class|0DEBXAJ5SA]] of $a$ and $C_G(a)$ is its [[centraliser|02YYV36BFM]].\nIn particular, if $G$ is finite, then $|\\operatorname{Cl}(a)|$ divides $|G|$.\n\n## Remarks\n\nThis is the [[orbit-stabiliser theorem|0E2HSCKZ6C]] for the [[action by conjugation|0P8CY9V437]].\nDirectly, $gag^{-1} = hah^{-1}$ if and only if $h^{-1}g \\in C_G(a)$.\nFor example, in $S_3$ the centraliser of $(1\\ 2\\ 3)$ is $A_3$, of index $2$, and the class of $(1\\ 2\\ 3)$ has $2$ elements.\n\n## Explanations\n\n### L1 human\n\nConjugates of $a$ by two elements are equal exactly when the two elements differ by something that commutes with $a$.\nSo the conjugates of $a$ correspond to the cosets of the centraliser.\nAn element that commutes with many elements has few conjugates.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "kind": "statement",
    "msc": [
      "20E45"
    ],
    "name": "Size of a conjugacy class is the index of the centraliser",
    "role": "proposition",
    "sources": [
      {
        "locator": "Conjugation as a group action, centralizers, and the class equation",
        "url": "https://en.wikipedia.org/wiki/Conjugacy_class"
      }
    ],
    "tag": "0VK5V9P8G3",
    "uses": [
      "0Q88KQD17A",
      "0Q0Z1BEXTF",
      "0E2HSCKZ6C",
      "0P8CY9V437"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0VK5V9P8G3/",
  "links": [
    {
      "field": "uses",
      "target": "0Q88KQD17A"
    },
    {
      "field": "uses",
      "target": "0Q0Z1BEXTF"
    },
    {
      "field": "uses",
      "target": "0E2HSCKZ6C"
    },
    {
      "field": "uses",
      "target": "0P8CY9V437"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0VK5V9P8G3.md",
  "msc": [
    "20E45"
  ],
  "name": "Size of a conjugacy class is the index of the centraliser",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0VK/0VK5V9P8G3.md",
  "status": null,
  "tag": "0VK5V9P8G3",
  "uri": "m2://0VK5V9P8G3",
  "url": "https://math2.tamnd.com/wiki/size-of-a-conjugacy-class-is-the-index-of-the-centraliser/",
  "writing": {
    "tag": "0VK5V9P8G3",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}