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.

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About Anurag Bishnoi

Currently a maths PhD student at Ghent University working in the Incidence Geometry research group. I am broadly interested in combinatorics, finite geometry and group theory.
This entry was posted in Finite Geometry. Bookmark the permalink.

3 Responses to Introduction to finite geometry I

  1. Signify says:

    Left maths a long time ago….maybe this will rejuvenate some of it.

  2. And, I have my first follower. Thank you 🙂

    I’ll be posting the second part soon. You can ask if something is unclear.

  3. Pingback: My first publication | Anurag's Math Blog

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