The general structure of the projective planes admitting $\mathrm{PSL}(2,q)$ as a collineation group

Abstract

Projective planes of order $n$ admitting $PSL\left(2,q\right)$, $q>3$, as a collineation group are investigated for $n\le q^2$. As a consequence, affine planes of order $n$ admitting $PSL\left(2,q\right)$, $q>3$, as a collineation group are classified for $n<q^2$ and $\left(q,n\right)\ne \left(5,16\right)$. Finally, a complete classification of the translation planes order $n$ that admitting $PSL\left(2,q\right)$, $q>3$, as a collineation group is obtained for $n\le q^2$.