What is the characteristic of a projective plane?

Question

Let $p \in \mathbb{N}$ be a prime number and $E$ the projective plane induced by the one and two dimensional linear subspaces of $(\mathbb{F}_p)^3$. I shall prove, that the characteristic of $E$ is equal to $p$. However I dont know what is meant by the characteristic of a projective plane and how it is defined. Can't find it anywhere.

Answer 1

OP has clarified that what was meant was the order of the projective plane, but hasn't indicated what definition of order has been given in class. The order is one less than the number of points on a line, but I don't know whether that's the definition of order or a theorem that follows from some other definition, and I don't know whether OP is allowed to use this. Anyway, let's show that the number of points on a line is $p+1$ (and therefore the order is $p$).

Remember that in this context the lines are 2-dimensional subspaces of a 3-dimensional vector space over a field of $p$ elements, and the points are the 1-dimensional subspaces. So we want to show that any 2-dimensional subspace contains $p+1$ 1-dimensional subspaces.

A 1-dimensional subspace is just a non-zero vector and all of its multiples by elements of the base field, so it contains $p-1$ nonzero vectors. A 2-dimensional subspace contains $p^2$ vectors, of which $p^2-1=(p-1)(p+1)$ are nonzero. No two distinct 1-dimensional subspaces have a nonzero vector in common, and every vector in the 2-dimensional subspace is in some 1-dimensional subspace, so there must be $p+1$ 1-dimensional subspaces contained in the 2-dimensional subspace, as was to be proved.