Does the projective closure of an affine plane of order $n$ also have order $n$?
I know that there exists a projective plane of order $n$ if and only if there exists an affine plane of order $n$, although I have not seen the proof of this statement. I am pretty sure the answer to the question above is yes, but I would like to know why.
Thank you!
Yes. The order of a block design is the number of blocks through a point minus the numbef of blocks containing any pair of points. In an affine plane of order $n$, each point lies on $n+1$ lines, and any pair of points lie together on one line. When you create the projective closure, every point still lies on $n+1$ lines, and there is still a single line through every pair of points.
The concept of a projective closure is half of the proof that an affine plane of order $n$ exists iff a projective plane of order $n$ exists. The other direction requires you to start with a projective plane and remove one line and the points on it. Showing the axioms for an affine plane is not hard, and I encourage you to give it a try!