Let the projective line $\mathbb{P_1}$ over the field of $\mathbb{Z/2Z}$. I'm asked to prove there are 6 projectivities between $\mathbb{P_1} \longrightarrow \mathbb{P_1}$ and giving the equations.
This is what I thought about it. As long as there is 2 classes in $\mathbb{Z/2Z}$ and four points to determine the projectivity, we can combine them in 7 ways, acording to the condition $ad - bc \neq 0$: $(0,1,1,1)$ ,$(1,0,1,1)$, $(1,1,0,1)$, $(1,1,1,0)$, $(0,1,1,0)$, $(1,0,0,1)$. Because if we take $(1,1,1,1)$:
$$z \rightarrow \frac{z+1}{z+1} = 1$$ which is no line. Actually, all equations goes by:
$$z \rightarrow \frac{az+b}{dz+d}$$
I'm not sure to need to prove there are projectivities, that can be done through the double reason. Is it correct?
Edit:
Let $T: E \longrightarrow E'$ an isomorphism between $\mathbb{K}$- vectorial spaces. The aplication $\hat{T}: \mathbb{P}(E) \longrightarrow \mathbb{P}(E')$ and $\pi: E \longrightarrow \mathbb{P}(E) $, then $\hat{T}$ is projectivity iff $\hat{T}(\pi (e)) = \pi (T(e))$.