{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\Pi$ be a [[projective plane|DANFNN8C4Y]] with a finite set of points.\nThe *order* of $\\Pi$ is the integer $n$ such that one line of $\\Pi$, and then every line, contains exactly $n + 1$ points.\n\n## Remarks\n\nThe definition does not depend on the line.\nGiven two distinct lines $\\ell$ and $m$, choose a point $O$ on neither; the map $P \\mapsto OP \\cap m$ is a bijection from the points of $\\ell$ to the points of $m$.\nFor a point $P$ and a line $\\ell$ not through $P$, the map $Q \\mapsto PQ$ is a bijection from the points of $\\ell$ to the lines through $P$.\nSo every point lies on exactly $n + 1$ lines.\nSince a projective plane has four points with no three on a line, every line has at least three points, so $n \\ge 2$.\n\nExample: in the [[Fano plane|QQ8GW1ASZM]] each line has $3$ points, so its order is $2$.\nIn $PG(2, q)$, a line is a $2$-dimensional subspace of $\\F_q^3$, which contains $(q^2 - 1)/(q - 1) = q + 1$ one-dimensional subspaces; so the order is $q$.\n\nNon-example: the real projective plane has infinitely many points on each line, so it has no order in this sense.\n\nIn Mathlib the order of a finite projective plane is `Configuration.ProjectivePlane.order`, defined through the number of lines through a point, which gives the same number.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "K2YVMTGTBB",
    "kind": "definition",
    "msc": [
      "51E15",
      "05B25"
    ],
    "name": "Order of a projective plane",
    "sources": [
      {
        "locator": "Finite projective planes",
        "url": "https://en.wikipedia.org/wiki/Projective_plane"
      }
    ],
    "tag": "D2GQNNQ7FR",
    "uses": [
      "DANFNN8C4Y",
      "2HWB7P1DZN",
      "QQ8GW1ASZM"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/D2GQNNQ7FR/",
  "links": [
    {
      "field": "defines",
      "target": "K2YVMTGTBB"
    },
    {
      "field": "uses",
      "target": "DANFNN8C4Y"
    },
    {
      "field": "uses",
      "target": "2HWB7P1DZN"
    },
    {
      "field": "uses",
      "target": "QQ8GW1ASZM"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/D2GQNNQ7FR.md",
  "msc": [
    "51E15",
    "05B25"
  ],
  "name": "Order of a projective plane",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/D2G/D2GQNNQ7FR.md",
  "status": null,
  "tag": "D2GQNNQ7FR",
  "uri": "m2://D2GQNNQ7FR",
  "url": "https://tamnd.github.io/math2/wiki/order-of-a-projective-plane-definition/",
  "writing": {
    "tag": "D2GQNNQ7FR",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}