Jump to content

User:Jeremydover/sandbox/Desarguesian plane of order 3

From Wikipedia, the free encyclopedia
Projective plane of order 3
Order3
Lenz-Barlotti ClassVII.2
Automorphisms
Point Orbit Lengths13
Line Orbit Lengths13
PropertiesDesarguesian
Self-dual

Construction

[edit]

This plane can be constructed using the standard vector space construction with a three-dimensional vector space over the finite field GF(3).

Ovals and Hyperovals

[edit]

This plane has odd order, so by Qvist's theorem it does not have a hyperoval. Being a Desarguesian, this plane does have ovals, and by Segré's theorem, all ovals in this plane are conics, being the set of absolute points of an orthogonal polarity.

Subplanes

[edit]

Since this plane is a Desarguesian plane of prime order, it has no proper subplanes.

Unitals

[edit]

Since the order of this plane is not square, it does not have a unital.