Question in geometry on Fano Plane

Question

Hello friends I have a geometry homework question asking me to do the following: I need to prove all projective planes of order two are isomorphic by showing they are all isomorphic to the Fano Plane. Any help would be appreciated Thanks all

Answer 1

It seems as if you can demonstrate that any such plane is the Fano plane by exhaustion.

By duality, every point has exactly three lines going through it. Let $L$ be a line and $P$ be a point not on $L$. The set of lines connecting a point on $L$ with $P$ exhausts the lines through $P$. It's not hard to show that at this point you already have $7$ points.

Now there are $21$ sets of order $2$ that you can create from these $7$ points which can be thought of as the lines they determine; however, since each line has three points on it, there are three different ways to choose two points from those three that all produce the same line, so in fact there are only $7$ lines after this redundancy is removed.

Conforming to the axioms of projective geometry, there is only one way to draw these $7$ lines with $7$ points.

Another way: coordinatize the plane with $\Bbb F$ in homogeneous coordinate triples. We know $\Bbb F$ has to have order $2$ since the plane is of order $2$. There are $2^3-1$ nonzero homogenous coordinate triples, corresponding with $7$ points. By duality again, you have $7$ lines.