Math 2.0

Geometry Corollary

Finite Desarguesian planes are Pappian

Builds on Wedderburn's little theoremDesarguesian planePappus's hexagon theoremDesargues's theoremDivision ringFinite fieldProjective planeCoordinatization of projective planes

Statement

Let Π\Pi be a finite projective plane in which Desargues's theorem holds. Then Pappus's hexagon theorem holds in Π\Pi. Moreover Π\Pi is isomorphic to PG(2,q)PG(2, q), the projective plane over the finite field with qq elements, for a prime power qq.

Remarks

A Desarguesian plane is coordinatized by a division ring DD, in the sense that Π≅PG(2,D)\Pi \cong PG(2, D); see the coordinatization of projective planes. If Π\Pi is finite, DD is finite, so DD is a field by Wedderburn's little theorem. A finite field has prime power order qq, and in a plane over a field Pappus's theorem holds.

Finiteness is necessary. The projective plane over the quaternions is Desarguesian, but Pappus's theorem fails in it, because the quaternions are not commutative.

In the other direction, Hessenberg's theorem says that Pappus's theorem implies Desargues's theorem in every projective plane. So for finite planes the two configuration theorems are equivalent.

Going deeper

For a student

Desargues's theorem lets one build coordinates from a division ring, and Pappus's theorem holds exactly when that division ring is commutative. Wedderburn's little theorem is therefore the algebraic input that makes every finite Desarguesian plane classical. It also shows that the order of a finite Desarguesian plane is a prime power. A finite plane that is not of the form PG(2,q)PG(2, q), such as the Hall plane of order 99, must therefore fail Desargues's theorem.

Reading path

To understand this page from the ground up, read these 20 pages in order. The foundations come first.

Show the path as a graph
Collinear points, concept, no evidence levelCollinear pointsconceptDesarguesian plane, concept, no evidence levelDesarguesian planeconceptDivision ring, concept, no evidence levelDivision ringconceptField, concept, no evidence levelFieldconceptFinite field, concept, no evidence levelFinite fieldconceptGroup, concept, no evidence levelGroupconceptIncidence structure, concept, no evidence levelIncidence structureconceptPlanar ternary ring, concept, no evidence levelPlanar ternary ringconceptProjective plane, concept, no evidence levelProjective planeconceptProjective space, concept, no evidence levelProjective spaceconceptPappus's hexagon theorem, statement, no evidence levelPappus's hexagon theo…statementRing, definition, no evidence levelRingdefinitionDesargues's theorem, statement, no evidence levelDesargues's theoremstatementCoordinatization of projective planes, statement, no evidence levelCoordinatization of p…statementCommutative ring, definition, no evidence levelCommutative ringdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionField, definition, no evidence levelFielddefinitionDivision ring, definition, no evidence levelDivision ringdefinitionWedderburn's little theorem, statement, no evidence levelWedderburn's little t…statementFinite Desarguesian planes are Pappian, statement, no evidence levelFinite Desarguesian p…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Collinear points concept
  2. Desarguesian plane concept
  3. Division ring concept
  4. Field concept
  5. Finite field concept
  6. Group concept
  7. Incidence structure concept
  8. Planar ternary ring concept
  9. Projective plane concept
  10. Projective space concept
  11. Pappus's hexagon theorem statement
  12. Ring definition
  13. Desargues's theorem statement
  14. Coordinatization of projective planes statement
  15. Commutative ring definition
  16. Unit of a ring definition
  17. Field definition
  18. Division ring definition
  19. Wedderburn's little theorem statement
  20. Finite Desarguesian planes are Pappian statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Wedderburn%27s_little_theorem
About this page
Tag6QBGAKFB9N
URIm2://6QBGAKFB9N
Kindstatement, corollary
WritingW0 stub
MSC51A30, 51E15, 12E15
StatusThis copy of the site has no build data. Run m2 status 6QBGAKFB9N.
AboutDesarguesian plane, Pappus's hexagon theorem
UsesWedderburn's little theorem, Desarguesian plane, Pappus's hexagon theorem, Desargues's theorem, Division ring, Finite field, Projective plane, Coordinatization of projective planes

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