On the geometry of {\textsf{O'N}}
A \(BCDL_{2003}\)-geometry is a coset geometry which is firm, residually connected, flag-transitive, residually weakly primitive, locally two-transitive and satisfies \((IP)_2\), the intersection property on all its rank two residues. \textit{F. Buekenhout} [Contemp. Math. 45, 1--32 (1985; Zbl 0577.51011)] gave a flag-transitive \(BCDL_{2003}\)-geometry of rank 4 for O'Nan's sporadic simple group O'N. Furthermore, \textit{F. Buekenhout} and the author [J. Comb. Theory, Ser. A 85, 148--164 (1999; Zbl 0930.20014)] showed that O'N has no residually weakly primitive geometry that satisfies \((IP)_2\) such that its rank is at least 5 with some maximal parabolic subgroup isomorphic to the Janko group \(J_1\) or such that its rank is at least 6 with some maximal parabolic subgroup isomorphic to the Mathieu group \(M_{11}\). On the other hand, \textit{A. A. Ivanov} and \textit{S. V. Shpectorov} [Russ. Math. Surv. 41, 211--212 (1986); translation from Usp. Mat. Nauk 41, No. 3(249), 183--184 (1986; Zbl 0626.20009)] obtained a geometry of rank 5 that admits a flag-transitive action of O'N. However, this geometry is not residually weakly primitive. In the paper under review the author considers O'N and its geometries. In order to classify rank two \(BCDL_{2003}\)-geometries for O'N, he determines those pairs of subgroups \((G_0, G_1)\) that together generate O'N and such that \(G_0\) is a maximal subgroup of O'N, \(G_{01} := G_0 \cap G_1\) is maximal in both \(G_0\) and \(G_1\), and the action of \(G_i\), \(i = 0, 1\), on the cosets of \(G_{01}\) in \(G_i\) is two-transitive. There are, up to isomorphism, nine maximal subgroups of O'N each of which may have several candidates for \(G_{01}\). A case-by-case analysis then shows that, up to isomorphism, there are ten rank two \(BCDL_{2003}\)-geometries for O'N. The author further proves that a \(BCDL_{2003}\)-geometry for O'N can have rank at most 4. Of course, this bound is sharp as the geometry due to Buekenhout [loc. cit.] shows. Using MAGMA, the author obtains that for four of the maximal subgroups of O'N found before, the maximal rank is 3 (so that none of these subgroups can be used as a maximal parabolic subgroup) and that for the remaining five maximal subgroups it is 4. For these latter subgroups, it is then checked case by case that none of their rank four geometries can be extended to a rank five geometry in O'N.
scientific article; zbMATH DE number 5808437
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