{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nA [[projective plane|DANFNN8C4Y]] is a *Moufang plane* if it satisfies the little Desargues property for every line $\\ell$ and every point $O$ on $\\ell$:\n\n1. let $ABC$ and $A'B'C'$ be triangles with no vertex on $\\ell$, with $A \\ne A'$, $B \\ne B'$, $C \\ne C'$, and with $AA'$, $BB'$, $CC'$ through $O$;\n2. suppose the points $AB \\cap A'B'$ and $AC \\cap A'C'$ lie on $\\ell$;\n3. then the point $BC \\cap B'C'$ also lies on $\\ell$.\n\n## Remarks\n\nSo three [[collinear|0AFNNTRTJE]] intersection points are required only when the centre $O$ lies on the axis $\\ell$.\nEquivalently, every line of the plane is a translation line, that is, the group of elations with axis $\\ell$ acts transitively on the points off $\\ell$.\nA projective plane is a Moufang plane exactly when it can be coordinatized by an alternative division ring.\n\nExample: $\\mathbb{P}^2(k)$ over a field $k$ is a Moufang plane.\nCheck: Desargues's theorem holds in $\\mathbb{P}^2(k)$ for all triangles in perspective, in particular when the centre lies on the axis.\n\nBy the Artin–Zorn theorem every finite alternative division ring is a field, so every finite Moufang plane is Desarguesian.\n\nNon-example: the Hall plane of order $9$ is a finite [[non-Desarguesian plane|HYE5FT4M34]].\nSo by the previous sentence it is not a Moufang plane: the little Desargues property fails in it for some line $\\ell$ and point $O$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "7F4HSNJJH8",
    "kind": "definition",
    "msc": [
      "51A35",
      "17D05"
    ],
    "name": "Moufang plane",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Moufang_plane"
      }
    ],
    "tag": "P5HCD49REV",
    "uses": [
      "DANFNN8C4Y",
      "0AFNNTRTJE"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/P5HCD49REV/",
  "links": [
    {
      "field": "defines",
      "target": "7F4HSNJJH8"
    },
    {
      "field": "uses",
      "target": "DANFNN8C4Y"
    },
    {
      "field": "uses",
      "target": "0AFNNTRTJE"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/P5HCD49REV.md",
  "msc": [
    "51A35",
    "17D05"
  ],
  "name": "Moufang plane",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/P5H/P5HCD49REV.md",
  "status": null,
  "tag": "P5HCD49REV",
  "uri": "m2://P5HCD49REV",
  "url": "https://tamnd.github.io/math2/wiki/moufang-plane-definition/",
  "writing": {
    "tag": "P5HCD49REV",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}