{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Example\n\nThe symmetric group $S_3$ has the six elements $e$, $(1\\ 2)$, $(1\\ 3)$, $(2\\ 3)$, $(1\\ 2\\ 3)$ and $(1\\ 3\\ 2)$.\nIt is [[isomorphic|08MG7H8JW5]] to the [[dihedral group|0JZ27DMJ5D]] $D_3$ of the symmetries of an equilateral triangle, when you number the vertices $1$, $2$, $3$.\nThe $3$-cycles are the rotations, and the transpositions are the reflections.\nIt is not [[abelian|082GHNPJ53]], because $(1\\ 2)(2\\ 3) = (1\\ 2\\ 3)$ but $(2\\ 3)(1\\ 2) = (1\\ 3\\ 2)$.\nThus $S_3$ is the smallest group that is not abelian.\nIts [[conjugacy classes|0DXD4QSZ8A]] are $\\{e\\}$, the three transpositions, and the two $3$-cycles.\nIts subgroups are $\\{e\\}$, three subgroups of order $2$, the [[alternating group|0Z5905Q73D]] $A_3$ of order $3$, and $S_3$.\nOnly $\\{e\\}$, $A_3$ and $S_3$ are [[normal|0CB0DX5BVV]].\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "instance_of": [
      "0515A6D2Y1",
      "0JZ27DMJ5D"
    ],
    "kind": "example",
    "msc": [
      "20B30"
    ],
    "name": "The symmetric group S3",
    "same_as": {
      "wikidata": "Q2204630",
      "wikipedia": "Dihedral group of order 6"
    },
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Dihedral_group_of_order_6"
      },
      {
        "locator": "Permutations of a set of three objects",
        "url": "https://en.wikipedia.org/wiki/Dihedral_group_of_order_6"
      },
      {
        "locator": "Conjugacy classes",
        "url": "https://en.wikipedia.org/wiki/Dihedral_group_of_order_6"
      }
    ],
    "tag": "09H9Y0Q11W",
    "uses": [
      "0RJMNEN3BJ",
      "09GK3DE0WV",
      "0D1VWQGZZ7"
    ]
  },
  "kind": "example",
  "link": "https://math2.tamnd.com/node/09H9Y0Q11W/",
  "links": [
    {
      "field": "instance_of",
      "target": "0515A6D2Y1"
    },
    {
      "field": "instance_of",
      "target": "0JZ27DMJ5D"
    },
    {
      "field": "uses",
      "target": "0RJMNEN3BJ"
    },
    {
      "field": "uses",
      "target": "09GK3DE0WV"
    },
    {
      "field": "uses",
      "target": "0D1VWQGZZ7"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/09H9Y0Q11W.md",
  "msc": [
    "20B30"
  ],
  "name": "The symmetric group S3",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/09H/09H9Y0Q11W.md",
  "status": null,
  "tag": "09H9Y0Q11W",
  "uri": "m2://09H9Y0Q11W",
  "url": "https://math2.tamnd.com/wiki/the-symmetric-group-s3/",
  "writing": {
    "tag": "09H9Y0Q11W",
    "kind": "example",
    "level": "W2",
    "next": [
      {
        "what": "a person confirmed each agent-written section",
        "note": "rule P04",
        "have": 0,
        "want": 1
      }
    ]
  }
}