I would like to begin a series of posts on what I and others might regard as the most important problems in “Finite Geometry”. I won’t attempt to give a complete survey, but I will attempt to highlight the most significant steps towards the problem. To begin, who could go past…
The Prime Power Conjecture
A projective plane is a finite incidence structure of points and lines satisfying the following axioms:
- every two points lie on a unique line,
- every two lines meet in a unique point.
We may also require a “non-degeneracy” condition, such as, “there exist four points with no three lying on a line”. The simplest example of a projective plane is the Fano plane:
Notice that the Fano plane has seven points and seven lines (the circle counts as a line). Now suppose we have two distinct lines and . By the axioms above, we know that and meet in a point . Consider a point which is not on both lines. Then every point on gives us a unique line on both and ; we can call this line . This line must meet in a unique point, and this point cannot be (exercise: why not?). So we have a bijection between the points of minus , and the points of minus ; specifically, this map is . So we see that every line of a projective plane contains the same number of points. We define the order of a projective plane to be one less than the number of points on any line. So we see that the Fano plane has order 2. In fact, if we take the one-dimensional subspaces of a 3-dimensional vector space over a finite field GF(q) to be the points and if we take the two-dimensional subspaces to be the lines, we have an example of a projective plane, which we call PG(2, q). So we have for every prime power q a projective plane of order q. In fact, there are four projective planes of order 9, three of which are not equivalent to PG(2,9), and these are examples of non-Desarguesian planes.
Now we are ready to state the conjecture:
Conjecture: The order of any finite projective plane is the power of a prime number.
The best result we have so far is the Bruck-Ryser-Chowla Theorem (1949-1950):
If a finite projective plane of order q exists and q is congruent to 1 or 2 (mod 4), then q must be the sum of two squares.
So there are no projective planes of orders 6 or 14. Tarry (1901) proved in his paper “Le problème des 36 ofﬁciers” that there is no finite projective plane of order 6, and a controversial computer-aided proof was given by Lam (1988) for the non-existence of a plane of order 10. There is much much more on the prime power conjecture, so we will leave you now to discover more; there are plenty of articles about and they are easy to find on the web…
Now for a survey (I can’t resist!):