Projective Planes with a Doubly Transitive Projective Subplane

Transitive projective planes

A long-standing conjecture is that any transitive finite projective plane is Desarguesian. We make a contribution towards a proof of this conjecture by showing that a group acting transitively on the points of a non-Desarguesian projective plane must not contain any components. 1 Background definitions and main results We say that a projective plane is transitive (respectively primitive) if it ...

Transitive projective planes and insoluble groups

Suppose that a group G acts transitively on the points of P, a finite non-Desarguesian projective plane. We prove first that the Sylow 2subgroups of G are cyclic or generalized quaternion; we then prove that if G is insoluble then G/O(G) is isomorphic to SL2(5) or SL2(5).2. MSC(2000): 20B25, 51A35.

Transitive hyperovals in finite projective planes

Transitive projective planes and 2-rank

Suppose that a group G acts transitively on the points of a nonDesarguesian plane, P. We prove first that the Sylow 2-subgroups of G are cyclic or generalized quaternion. We also prove that P must admit an odd order automorphism group which acts transitively on the set of points of P. 1 MSC(2000): 20B25, 51A35.

Smooth Projective Planes

Using symplectic topology and the Radon transform, we prove that smooth 4-dimensional projective planes are diffeomorphic to CP. We define the notion of a plane curve in a smooth projective plane, show that plane curves in high dimensional regular planes are lines, prove that homeomorphisms preserving plane curves are smooth collineations, and prove a variety of results analogous to the theory ...