---
tag: NH7S9CHHBN
kind: definition
name: Finite geometry
msc: ["51E99", "05B25"]
defines: MCDZP31ZF3
uses: [FGKNAFNHA3, DANFNN8C4Y, EC7XT2ZM5S, ZT9A6JK8QF]
sources:
  - url: https://en.wikipedia.org/wiki/Finite_geometry
---

## Definition

A *finite geometry* is an [[incidence structure|FGKNAFNHA3]] $(P, L, I)$ with a finite set $P$ of points that satisfies the axioms of a given type of geometry.
The usual types are [[projective planes|DANFNN8C4Y]], [[affine planes|EC7XT2ZM5S]], and projective and affine spaces of higher dimension.

## Remarks

Example: the [[projective space|ZT9A6JK8QF]] $PG(2, 2)$ over the field with two elements, the [[Fano plane|QQ8GW1ASZM]].
It has $7$ points and $7$ lines.
Any two points lie on exactly one line, and any two lines meet in exactly one point.
The points $[1 : 0 : 0]$, $[0 : 1 : 0]$, $[0 : 0 : 1]$, $[1 : 1 : 1]$ have no three on a line, so it is a finite projective plane.

Non-example: the real projective plane satisfies the axioms of a projective plane, but it has infinitely many points, so it is not a finite geometry.

The term also names the field of mathematics that studies these structures.
Finite projective spaces of dimension at least $3$ all come from finite fields, by the [[Veblen–Young theorem|NPNNXWMVCC]] and Wedderburn's little theorem; finite planes need not.