Biplanes with flag-transitive automorphism groups of almost simple type, with alternating or sporadic socle.
Biplanes are a generalization of projective planes in the sense that it is assumed that there are exactly two lines through any two points; in design theory language, a biplane is a \((v,k,2)\)-symmetric design. Very few are known, there are none known for \(k>13\). This paper contributes to the problem by considering biplanes with flag-transitive automorphism groups. The corollary of the main theorem is: Let \(D\) be a biplane with a flag-transitive automorphism group \(G\), then one of the following holds: (1) \(D\) has parameters \((16,6,2)\), (2) \(G\leq A\Gamma L_1(q)\) for some odd prime power, or (3) \(G\) is almost simple with socle a Chevalley group. The proof depends on the fact that we have a complete description of the maximal subgroups and hence we can identify the possible stabilizers of points quite easily. In the sporadic case the results come from knowing the maximal subgroups and largely using numerology to show that there is no such action.
scientific article; zbMATH DE number 2165401
scientific article