For a finite field $\mathbb{F}$, the projective general linear group $ \text{PGL}_2(\mathbb{F}) $ is a group of linear transforms $\mathbb{F}\cup \{\infty \}\to \mathbb{F}\cup \{\infty \}$
$$ u\to \frac{\text{$\alpha $u}+\beta }{\text{$\gamma $u}+\delta } $$
Projective semilinear group $\text{P$\Gamma $L}_2(\mathbb{F}) $ is an extension $ \text{PGL}_2(\mathbb{F}) $ by the Frobenius endomorphisms $\rho :\mathbb{F}\to \mathbb{F} $
However I dont know whether the extension contains all of the transformations
$$ u\to \frac{\alpha \rho _1 (u) + \beta }{\gamma \rho _2 (u)+ \delta } $$
or them with $ \rho _1 (u) = \rho _2 (u) $ only.
Or both definitions are equivalent?
Best Answer
A curious question that I'm a bit ashamed I had to think about for a while:-)
I am convinced that you can only include the transformations with $\rho_1=\rho_2$. At least two things will go wrong otherwise. In my examples $F$ stands for the Frobenius automorphism $F(x)=x^p$.