Translation planes of large dimension admitting nonsolvable groups
Authors look for (finite) translation planes with large Ostrom dimension admitting nonsolvable collineation groups. Using the technique of spreads and irreducible pairs (representation of a translation plane in vector space over its kernel) the authors show that the spread of order \(q^ 3\) associated with an irreducible pair on an \(3m\)-dimensional vector space over \(GF(p)\), \(q=p^ m\), admits \(GL(2,q)\) with sufficiently regular action [Cor. (4.5)]. Explicit examples of irreducible pairs [Cor. (6.2) and Thm. (6.3)] yield the main result: for any integer \(N\) there exist a translation plane of dimension \(>N\) and order \(q^ 3\) admitting \(GL(2,q)\) as a collineation group. Known examples of planes with small kernels, due to Kantor and to Bartolone and Ostrom can be obtained following the technique proposed in the paper.
scientific article; zbMATH DE number 98731
scientific article