Translation planes of odd order and odd dimension

Point-primitive Inversive Planes of Odd Order

A famous but still unsolved problem in finite geometries is the question of whether a finite inversive plane J of odd order n is necessarily miquelian, that is, it arises from the plane sections of an elliptic quadric in PG(3,«). The question has been approached from different angles, and it has a positive answer if suitable conditions are added to J. The most classical result in this context i...

3 ODD ORDER PLANES 3 Result

Groups of generalised projectivities in projective planes of odd order

The generalised projectivities (GP's) associated with projective planes of odd order are investigated. These are non-singular linear mappings over GF(2) defined from the binary codes of these planes. One case that is investigated in detail corresponds to the group of an affine plane-every point corresponds to a GP. It is shown how many collineations that fix the line at infinity point-wise can ...

Odd order flag-transitive a‰ne planes of dimension three over their kernel

With the exception of Hering’s plane of order 27, all known odd order flag-transitive a‰ne planes are one of two types: admitting a cyclic transitive action on the line at infinity, or admitting a transitive action on the line at infinity with two equal-sized cyclic orbits. In this paper we show that when the dimension over the kernel for these planes is three, then the known examples are the o...

Odd order flag-transitive affine planes of dimension three over their kernel

With the exception of Bering's plane of order 27, all known odd order flag-transitive affine planes are one of two types: admitting a cyclic transitive action on the line at infinity, or admitting a transitive action on the line at infinity with two equal-sized cyclic orbits. In this paper we show that when the dimension over the kernel for these planes is three, then the known examples are the...