Generalized quadrangles and their connection to prime powers.

Question

Generalized quadrangles are a commonly know geometric structures.

A generalized quadrangle is an incidence structure $(P,B,I)$, with $I \subseteq P \times B$ an incidence relation, satisfying certain axioms. Elements of $P$ are by definition the points of the generalized quadrangle, elements of $B$ the lines.

The axioms are the following:

• There is an $s$ ($s \geq 1$) such that on every line there are exactly $s + 1$ points. There is at most one point on two distinct lines.
• There is a $t$ ($t \geq 1$) such that through every point there are exactly $t + 1$ lines. There is at most one line through two distinct points.
• For every point $p$ not on a line $L$, there is a unique line $M$ and a unique point $q$, such that $p$ is on $M$, and $q$ on $M$ and $L$.

($s$,$t$) are the ''parameters'' of the generalized quadrangle.

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For some parameters the existence of GQs are known. But still only the following parameters have been found possible until now, with $q$ an arbitrary prime power:

• $(q,q)$
• $(q,q^2)$ and $ (q^2,q)$
• $(q-1,q+1)$ and $(q+1,q-1)$
• and $(q^2,q^3)$ and $(q^3,q^2)$

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My questions are:

Where comes the prime power from? Why are all currently known sets of parameters linked to prime powers (what is their advantage)?

It somehow seem arbitrary to me that prime powers work out better than other parameters (not prime powers).

Answer 1

The constructions for generalized quadrangles are based on the existence of either a finite field $\mathbb{F}_q$ of order $q$ or of a projective (or affine) space of order $q$.

It is known that a finite field of order $q$ exists if and only if the order $q$ is a prime power.

For affine space, it is an open conjecture

Every finite affine plane has order equal to a prime power

The problem is still open, meaning that we have no counter-example.