## Introduction to finite geometry I

We (most of us) are familiar with the Euclidean Geometry and the ${5}$ (really ${20}$) axioms it is based on. Let’s consider a simpler system here called Affine plane.

Affine Plane

Def : An Affine plane is a triple ${(P,L,I)}$ where ${P}$ is a set of points, ${L}$ a set of lines and ${I \subseteq P \times L}$ an incidence relation (just a fancy name) such that :

1. Each line is incident with at least two points and each point is incident with at least three lines.
2. Any two distinct points determine a unique line.
3. (Playfair axiom) Given any point ${x}$ and a line ${l}$, there exists a unique line ${m}$ incident to ${x}$ which is parallel to ${l}$.

Note that two lines ${l}$, ${m}$ are called parallel if either ${l = m}$ or there is no point incident to both ${l}$ and ${m}$. The first axiom simply rules out some trivial cases like ${P = \emptyset}$, ${L = \emptyset}$, ${L = \{l\}}$ and all points incident to ${l}$, etc.

An alternate view here would be to consider ${L}$ as a class of subsets of ${P}$, then instead of the incidence relation we say ${x}$ belongs to ${l}$ or ${l}$ contains ${x}$ where ${x}$ is a point and ${l}$ is a line. See this for more details.

Lemma 1 Assuming the first two axioms to be true, playfair axiom is equivalent to the statement “The parallel relation (defined above) is an equivalence relation and each parallel class is a partition of ${P}$ in the sense that each point is incident with a unique line in a parallel class”.

The most common example of an Affine plane is the Euclidean plane ${{\mathbb R}^2}$. Here is another example:

Let ${F}$ be a division ring. Consider the system called ${AG(2,F)}$ whose point set is ${ F \times F }$ and lines are all sets of the form ${\{ (x,y) \in F \times F ~:~ x = c\}}$ and ${\{ (x,y) \in F \times F ~:~ y = mx + c\}}$ for ${m,c \in F}$.

It is an easy exercise to see that this defines an Affine plane. Things start getting interesting when we consider the possibility where set ${P}$ is finite. Here’s a non trivial theorem for the case of finite Affine plane:

Theorem 2 Let ${(P,L,I)}$ be a finite Affine plane, then there is a number ${n \geq 2}$ (called the order of the plane) such that :

1. Each line is incident with exactly ${n}$ points.
2. Each point is incident with exactly ${n+1}$ lines.
3. Each parallel class has ${n}$ lines.
4. There are ${n+1}$ parallel classes.
5. There are ${n^2}$ points.
6. There are ${n^2 + n}$ lines.

Proof: We’ll first show that any two points are incident to the same number of lines and call that number ${n+1}$. Let ${x}$, ${y}$ be two points. They determine a unique line which is incident to both, call it ${l}$. Now, because of the Playfair axiom, there is an obvious bijection between the lines incident to ${x}$ and lines incident to ${y}$ other than ${l}$. This proves 2.

Now, Let ${l}$ be a line and ${p}$ a point not incident to ${l}$ then there is a unique line ${m}$ parallel to ${l}$ and passing through ${p}$. Each of the other ${n}$ lines incident to ${p}$ intersect ${l}$ at a unique point. And corresponding to each point on ${l}$ we have a unique line determined by that point and ${p}$. Hence, the number of points on ${l}$ must be ${n}$.

Let ${l}$ be a line in a parallel class and ${p}$ a point incident to it. Let ${m}$ be a line different from ${l}$ and incident to ${p}$. There are ${n-1}$ other points on ${m}$. Through each of these points there is a line parallel to and distinct from ${l}$, and hence number of lines in the parallel class is at least ${n}$. Also, ${m}$ has a common point with all the members of this parallel class and hence the number of lines is at most ${n}$.

Let ${p}$ be a point. It has ${n+1}$ lines incident to it and all of them must lie in different parallel class. Also, ${p}$ is incident to a unique line in each parallel class. Hence, number of parallel classes must be ${n+1}$.

5. Follows from 1. and 3. while 6. follows from 3. and 4. Therefore, we have shown the existence of such an ${n}$. $\Box$

The simplest example of finite Affine plane is the structure with ${P = \{1,2,3,4\}}$ and ${L}$ = all ${2}$-subsets of ${P}$. Another one is the family of Affine planes given by ${AG(2, GF(q))}$ where ${GF(q)}$ is the Galois field of order ${q}$. Therefore, for any prime power ${q}$ we have a finite Affine plane of order ${q}$.

Given a finite affine plane, must its order be a prime power? Given a prime ${p}$, is there a unique affine plane of order ${p}$? These are two important open problems in the area of finite geometry.