Strongly irreducible collineation groups and the Hall-Janko group (Q1182538)

In [Abh. Math. Semin. Univ. Hamb. 49, 155-182 (1979; Zbl 0431.51002)] \textit{Chr. Hering} introduced the concept of a strongly irreducible collineation group on a projective plane. In [Geom. Dedicata 13, 7-46 (1982; Zbl 0496.51007)] \textit{A. Reifart} and the reviewer showed that besides \(J_ 2\) no sporadic simple group can act strongly irreducibly. In fact the group \(J_ 2\) acts strongly irreducibly on a quaternionic plane. It is conjectured that there is no finite projective plane admitting \(J_ 2\) as a strongly irreducible collineation group generated by perspectivities. In this paper the author investigates such a hypothetic plane and shows that its order \(n\) is a forth power of \(m\), where \(m\) is not divisible by 2, 3 or 5 and \(m\neq 7\). Some further results for \(m=11\) are obtained. The general question is still open.