R2-planes with point transitive 3-dimensional collineation group

Antiflag-transitive collineation groups revisited

An antiflag in a projective space is a non-incident point-hyperplane pair. A subgroup G of ΓL(n,q) is antiflag-transitive if it acts transitively on the set of antiflag of PG(n−1,q). In 1979, Cameron and Kantor [2] published a paper determining all antiflagtransitive subgroups of ΓL(n,q). A large part of the motivation was the fact that a group which acts 2-transitively on points is necessarily...

On Finite Point Transitive Affine Planes with Two Orbits

Finite flag-transitive affine planes with a solvable automorphism group

In this paper, we consider finite flag-transitive affine planes with a solvable automorphism group. Under a mild number-theoretic condition involving the order and dimension of the plane, the translation complement must contain a linear cyclic subgroup that either is transitive or has two equal-sized orbits on the line at infinity. We develop a new approach to the study of such planes by associ...

Projective Planes of Order 12 Do Not Have a Four Group as a Collineation Group

We have shown in [2] that the full collineation group of any projective plane of order 12 is a (2, 3) group. It is of interest to determine the structure of this (2,3} group. As a first step in that direction, we have shown in [3] that a non-Abelian group of order 6 cannot act as a collineation group on any projective plane of order 12. As a second step, we have shown in [4] that there is no pr...

On collineation groups of finite planes

From the Introduction to P. Dembowski’s Finite Geometries, Springer, Berlin 1968: “ . . . An alternative approach to the study of projective planes began with a paper by BAER 1942 in which the close relationship between Desargues’ theorem and the existence of central collineations was pointed out. Baer’s notion of (p, L)–transitivity, corresponding to this relationship, proved to be extremely f...