12.65 (1992)

Let $\mathscr{P} = (P_0, P_1, P_2)$ be a parabolic system in a finite group $G$, belonging to the $C_3$ Coxeter diagram $\underset{1}{\circ}\ \text{–––––} \underset{2}{\circ} \xlongequal{\quad\quad} \underset{3}{\circ}$ and let the Borel subgroup have index at least 3 in $P_0$ and $P_1$. It is known that, if we furthermore assume that the chamber system of $\mathscr{P}$ is geometric and that the projective planes arising as $\{0, 1\}$-residues from $\mathscr{P}$ are desarguesian, then either $G^{(\infty)}$ is a Chevalley group of type $C_3$ or $B_3$ or $G = A_7$. Can we obtain the same conclusion in general, without assuming the previous two hypothesis?