Element structure of general semilinear group of degree two over a finite field

This article describes the element structure of the general semilinear group of degree two over a finite field of size $$q = p^r$$, where $$p$$ is the characteristic of the field. The Galois group of the extension $$\mathbb{F}_q/\mathbb{F}_p$$ is a cyclic group of order $$r$$, generated by the $$p$$-power map (the Frobenius automorphism). Note that $$r = \log_p q$$