A rank 3 geometry for the O'Nan group connected with the Livingstone graph

Abstract

We construct a rank $3$ geometry $\mathrm{\Gamma }\left(O^{\prime }N\right)$ over the diagram $\begin{array} {lcr} \quad\mathrm{\small{C}} \qquad \small{8}\,\small{5}\,\small{8} \\ \circ ――\circ ――\circ \\ \small{1} \qquad \small{10} \qquad \small{1} \end{array}$ whose automorphism group is the O’Nan sporadic simple group. The maximal parabolic subgroups are the Janko group $J_1$, $2\times S_5$ and the Mathieu group $M_{11}$. Our construction is based on a convenient amalgam of known geometries of rank $2$ for $J_1$ and $M_{11}$ extracted from the subgroup lattice of $O^{\prime }N$.