{
  "aliases": [
    "parity-of-a-permutation"
  ],
  "backlinks": [
    {
      "field": "uses",
      "target": "0T13HM377F"
    }
  ],
  "body": "\n## Definition\n\nLet $n \\geq 2$ and let $\\sigma \\in S_n$.\nThe permutation $\\sigma$ is even when $\\operatorname{sgn}(\\sigma) = 1$, and odd when $\\operatorname{sgn}(\\sigma) = -1$.\n\n## Remarks\n\nThe identity is even and each [[transposition|0FWVRHBRWM]] is odd.\nA product of two even or two odd permutations is even, and a product of an even and an odd permutation is odd.\nFor $n \\geq 2$, $S_n$ has $n!/2$ even and $n!/2$ odd permutations.\nThe odd permutations do not form a subgroup, but they form a coset of $A_n$.\n\n## Explanations\n\n### L1 human\n\nEach permutation is even or odd, like each integer.\nEven and odd combine by the same rules as for integers.\nExactly half of the permutations are even.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "aliases": [
      "parity-of-a-permutation"
    ],
    "defines": "0SQTMQTSRZ",
    "kind": "definition",
    "msc": [
      "20B30"
    ],
    "name": "Even and odd permutations",
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Parity_of_a_permutation"
      },
      {
        "locator": "Properties",
        "url": "https://en.wikipedia.org/wiki/Parity_of_a_permutation"
      }
    ],
    "tag": "0771AJP4BX",
    "uses": [
      "0SK90238BH"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0771AJP4BX/",
  "links": [
    {
      "field": "defines",
      "target": "0SQTMQTSRZ"
    },
    {
      "field": "uses",
      "target": "0SK90238BH"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0771AJP4BX.md",
  "msc": [
    "20B30"
  ],
  "name": "Even and odd permutations",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/077/0771AJP4BX.md",
  "status": null,
  "tag": "0771AJP4BX",
  "uri": "m2://0771AJP4BX",
  "url": "https://math2.tamnd.com/wiki/even-and-odd-permutations-definition/",
  "writing": {
    "tag": "0771AJP4BX",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}