Four-dimensional compact projective planes with a nonsolvable automorphism group

Abstract

We contribute to the enumeration of all four-dimensiohal compact projective planes with an at least seven-dimensional automorphism group (cf. Betten [8]) by treating the nonsolvable case. Moreover, we find that the only possible six-dimensional nonsolvable automorphism group is N2. GL~-N.Geometriae Dedieata i36: [225][226][227][228][229][230][231][232][233][234] 1990.

4-dimensional compact projective planes with a 7-dimensional collineation group

“…Proof In [19] R. L6wen studies 4-dimensional compact projective planes where E is nonsolvable and has dimension at least 6. He proves that either E is isomorphic to the 6-dimensional group R 2" GL~-~ or the plane is a translation plane or the dual of a translation plane.…”

Actions of almost simple groups on compact eight-dimensional projective planes are classical, almost

“…The following has been proved, see [12], [15], and [26]; compare also [30, 33.6 The term 'action on a compact projective plane' will be defined precisely in 2.1 below. Our aim in the present note is to extend 1.…”

“…In fact, all actions of almost simple groups on 2-dimensional planes, and all actions of almost simple groups of dimension greater than 3 on 4-dimensional planes have been determined ( [22], [15]). In the present note, we determine all actions of almost simple groups of dimension greater than 10 on 8-dimensional planes.…”

4-Dimensional compact projective planes with small nilradical

“…The group E is known to be a Lie group with dim E < 16. All planes with dim ~ _> 7 are completely classified ( [4], [13]). All flexible translation planes (and hence also the dual translation planes) are classified ( [2]) and also all flexible shift planes ( [1], [11], [12], [3]).…”