On a geometry of Ivanov and Shpectorov for the O'Nan sporadic simple group (Q1284160)

The paper contributes to a project aiming to classify all geometries admitting flag-transitive actions of finite simple groups satisfying the properties \((\text{IP})_2\) (every rank 2 residue is either a partial linear space or a generalized digon) and RWPRI (for every flag its stabilizer acts primitively on the set of elements of at least one type in the corresponding residue). It is shown in the paper that for the O'Nan sporadic simple group \(O'N\) there are no \((\text{IP})_2\) and RWPRI geometries of rank 6 with \(M_{11}\) as a maximal parabolic or of rank 5 with \(J_1\) as a maximal parabolic. A geometry \({\mathcal G}(O'N)\) of rank 5 for \(O'N\) was constructed by \textit{A. A. Ivanov} and \textit{S. V. Shpektorov} [Usp. Mat. Nauk 41, No. 3(249), 183-184 (1986; Zbl 0626.20009)] (it has both \(M_{11}\) and \(J_1\) as maximal parabolics). By the result in the paper \({\mathcal G}(O'N)\) is not RWPRI. Up to the best of the reviewer's knowledge another important question about \({\mathcal G}(O'N)\) is unanswered: it is not known whether or not the triple cover of \({\mathcal G}(O'N)\) associated with the non-split extension of \(O'N\) by a centre of order 3 is universal.