{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0KFZ0QX83G"
    }
  ],
  "body": "\n## Statement\n\nLet $k \\geq 1$ and let $\\sigma$ be a $k$-cycle in $S_n$.\nThen $\\operatorname{sgn}(\\sigma) = (-1)^{k-1}$.\n\n## Remarks\n\nA $k$-cycle is a product of $k - 1$ transpositions, which gives the formula.\nThus a cycle is an [[even permutation|0SQTMQTSRZ]] exactly when its length is odd.\nWith the [[disjoint cycle decomposition|0YWDAZPRPQ]], the sign of $\\sigma \\in S_n$ is $(-1)^{n - c}$, where $c$ is the number of cycles, with the fixed points counted as $1$-cycles.\n\n## Explanations\n\n### L1 human\n\nA cycle of length $k$ is a chain of $k - 1$ swaps.\nSo a $3$-cycle is even and a transposition is odd.\nThe length and the parity of a cycle are opposite.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "kind": "statement",
    "msc": [
      "20B30"
    ],
    "name": "Sign of a cycle",
    "role": "proposition",
    "sources": [
      {
        "locator": "Properties",
        "url": "https://en.wikipedia.org/wiki/Parity_of_a_permutation"
      },
      {
        "locator": "Basic properties",
        "url": "https://en.wikipedia.org/wiki/Cyclic_permutation"
      }
    ],
    "tag": "0V0E04DWX2",
    "uses": [
      "0SK90238BH",
      "0XK6FECMW5"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0V0E04DWX2/",
  "links": [
    {
      "field": "uses",
      "target": "0SK90238BH"
    },
    {
      "field": "uses",
      "target": "0XK6FECMW5"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0V0E04DWX2.md",
  "msc": [
    "20B30"
  ],
  "name": "Sign of a cycle",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0V0/0V0E04DWX2.md",
  "status": null,
  "tag": "0V0E04DWX2",
  "uri": "m2://0V0E04DWX2",
  "url": "https://math2.tamnd.com/wiki/sign-of-a-cycle/",
  "writing": {
    "tag": "0V0E04DWX2",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}