On the geometry of the Hall-Janko group on 315 points
\textit{A. M. Cohen} [Lond. Math. Soc. Lect. Note Ser. 49, 81-87 (1981; Zbl 0472.05030)] introduced the geometry \({\mathcal G}\) on 315 points associated with the sporadic group HJ and described it in terms of quaternionic reflections. In the paper under review, \({\mathcal G}\) is now described as a subgeometry of the geometry \({\mathcal G}^*\) associated with the sporadic simple group Suz [cf. the author, J. Algebra 112, No.1, 198-239 (1988; Zbl 0635.51009)] fixed by some involutive automorphism. The points of \({\mathcal G}\) are being identified with a class of involutions in HJ, in which two points are collinear whenever the two corresponding involutions commute. The paper aims at deriving properties of \({\mathcal G}\) and HJ. In particular, the author succeeds in finding generators for the group of type 2.HJ, the suborbit-diagram of \({\mathcal G}\), and a classification and description of subgeometries of \({\mathcal G}\) with the structures of generalized n-gons. The explicit generators of the group of type 2.HJ are represented as 12\(\times 12\) matrices over \({\mathbb{C}}\) and as \(6\times 6\) matrices over \(F_ 9\).
scientific article; zbMATH DE number 4120890