Flag-transitive and almost simple orbits in finite projective planes

Abstract

Let $\mathrm{\Pi }$ be a projective plane of order $n$ and let $G$ be a collineation group of $\mathrm{\Pi }$ with a point-orbit $\mathcal{O}$ of length $v$. We investigate the triple $\left(\mathrm{\Pi },\mathcal{O},G\right)$ when $\mathcal{O}$ has the structure of a non trivial $2$-$\left(v,k,1\right)$ design, $G$ induces a flag-transitive and almost simple automorphism group on $\mathcal{O}$ and $n\le \sum \left(\mathcal{O}\right)=b+v+r+k$.