Projective linear actions and Iwasawa's hypotheses

The matrices $\left(\begin{array}{cc}1& b\\ 0& 1\end{array}\right)$ give translations $x\mapsto x+b$, so the stabilizer of $\mathrm{\infty }$ is transitive on ${\mathbb{F}}_q$. The matrix $\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$ sends $\mathrm{\infty }$ to $0$; composing it with translations sends $\mathrm{\infty }$ to any finite point. All these matrices have determinant one, so they act through $G$, not merely through the full projective linear group. Hence the action is transitive, and its point stabilizer is transitive on the other points: it is doubly transitive and therefore primitive by [L1]. A fractional linear map fixing $\mathrm{\infty }$, $0$, and $1$ is scalar as a matrix, hence the projective action is faithful.

The subgroup $A$ fixes $\mathrm{\infty }$, is abelian under composition, and is normal in the stabilizer of $\mathrm{\infty }$ because conjugating a translation by an affine map gives another translation.

Conjugating $A$ by the inversion $x\mapsto -1\mathrm{/}x$ gives the lower-unitriangular subgroup. The upper and lower unitriangular subgroups generate ${\mathrm{SL}}_2({\mathbb{F}}_q)$ by Gaussian elimination, so their projective images generate $G$. Thus the conjugates of $A$ generate $G$, and [L2] applies.