Flag-transitive point-primitive 2-(v,k,4) symmetric designs and two dimensional classical groups

Abstract

Let D be a 2-(v;k;4) symmetric design and G be a ∞ag-transitive point-primitive automorphism group of D with X E GAut(X) where X » PSL2(q). Then D is a 2-(15;8;4) symmetric design with X = PSL2(9) and Xx = PGL2(3) where x is a point of D. x1 Introduction A 2-(v;k;‚) design is an incidence structure D = (P;B) where P is a set of v points and B is a set of b blocks with an incidence relation such that every block is incident with exact k points, and every 2-element subset of P is incident with exact ‚ blocks. And D is called a symmetric design if v = b, and is nontrivial if ‚ < k < v i 1. An automorphism of a design D is a permutation of the points which also permutes the blocks. The set of automorphisms of a design with the composition of maps is a group. If G is a primitive permutation group on the point set P then G is called point-primitive, otherwise point-imprimitive. A ∞ag in a design is an incident point-block pair, G is called ∞ag-transitive if G is transitive on the set of ∞ags. There are many works on 2-(v;k;‚) symmetric designs with ‚ small, especially under the condition that the automorphism group of a symmetric design is ∞ag-transitive. For exam- ple, W. M. Kantor in (7) classifled ∞ag-transitive 2-(v;k;1) symmetric designs, which are called projective planes. In (11-14), Regueiro reduced the classiflcation of ∞ag-transitive 2-(v;k;2) sym- metric designs, i.e. biplanes, to the situation where the automorphism group is a 1-dimensional a-ne group. A 2-(v;k;3) symmetric design is called a triplane. In (18-21), Zhou and Dong proved that if D is a nontrivial triplane with a ∞ag-transitive automorphism group G, then D has parameters (11;6;3), (15;7;3), (45;12;3) or G is an a-ne group. Recently, they also clas- sifled the case that the automorphism group is of a-ne type. In (16), Praeger and Zhou have studied ∞ag-transitive, point-imprimitive 2-(v;k;‚) symmetric designs, especially for the cases that ‚ is at most 10. In 2009, Law, Praeger and Reichard classifled ∞ag-transitive (90;20;4)