{
  "aliases": [
    "vierergruppe"
  ],
  "backlinks": null,
  "body": "\n## Description\n\nThe Klein four-group $V$ is the group with four elements $e, a, b, c$, in which $e$ is the identity and each element is its own inverse.\nThe product of two distinct elements of $\\{a, b, c\\}$ is the third one.\nIt is [[abelian|082GHNPJ53]], and it is [[isomorphic|08MG7H8JW5]] to the [[direct product|00HF4RJ5QN]] $\\mathbb{Z}/2\\mathbb{Z} \\times \\mathbb{Z}/2\\mathbb{Z}$.\n\n## Remarks\n\nThe Klein four-group is the smallest group that is not [[cyclic|0KVGH0J5B3]], because it has no element of order $4$.\nIt is the symmetry group of a rectangle that is not a square.\nIn $S_4$, the set $\\{e, (1\\ 2)(3\\ 4), (1\\ 3)(2\\ 4), (1\\ 4)(2\\ 3)\\}$ is a normal subgroup isomorphic to $V$.\nEach group of order $4$ is isomorphic to $\\mathbb{Z}/4\\mathbb{Z}$ or to $V$.\n\n## Explanations\n\n### L1 human\n\nThe Klein four-group is like two light switches that you can flip separately.\nEach switch flipped two times is back to the start.\nFlipping both switches is the third element that is not the identity.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "aliases": [
      "vierergruppe"
    ],
    "kind": "object",
    "msc": [
      "20K01"
    ],
    "name": "Klein four-group",
    "same_as": {
      "wikidata": "Q550593",
      "wikipedia": "Klein four-group"
    },
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Klein_four-group"
      },
      {
        "locator": "Permutation representation",
        "url": "https://en.wikipedia.org/wiki/Klein_four-group"
      }
    ],
    "tag": "0PMRQJVTME",
    "uses": [
      "071DDFVG4J",
      "0YP7WYXF9V"
    ]
  },
  "kind": "object",
  "link": "https://math2.tamnd.com/node/0PMRQJVTME/",
  "links": [
    {
      "field": "uses",
      "target": "071DDFVG4J"
    },
    {
      "field": "uses",
      "target": "0YP7WYXF9V"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0PMRQJVTME.md",
  "msc": [
    "20K01"
  ],
  "name": "Klein four-group",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0PM/0PMRQJVTME.md",
  "status": null,
  "tag": "0PMRQJVTME",
  "uri": "m2://0PMRQJVTME",
  "url": "https://math2.tamnd.com/wiki/klein-four-group/",
  "writing": {
    "tag": "0PMRQJVTME",
    "kind": "object",
    "level": "W1",
    "next": [
      {
        "what": "section Invariants",
        "note": "a table of the main invariants",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge instance_of out",
        "have": 0,
        "want": 1
      }
    ]
  }
}