Subplanes of Order $3$ in Hughes Planes

Subplanes of Order 3 in Hughes Planes

L. Puccio and M. J. de Resmini [5] showed that subplanes of order 3 exist in the Hughes plane of order 25. (We refer always to the ordinary Hughes planes; equivalently, all our nearfields are regular.) Computations of the second author [2] show that among the known projective planes of order 25 (including 99 planes up to isomorphism/duality), exactly four have subplanes of order 3. These four p...

3 ODD ORDER PLANES 3 Result

Inversive Planes of Even Order

1. Results. An inversive plane is an incidence structure of points and circles satisfying the following axioms: I. Three distinct points are connected by exactly one circle. II. If P, Q are two points and c a circle through P but not Q, then there is exactly one circle c' through P and Q such that cC\c'~ [P]. III. There are at least two circles. Every circle has at least three points. For any p...

Transitive Arcs in Planes of Even Order

When one considers the hyperovals in PG (2 , q ) , q even , q . 2 , then the hyperoval in PG (2 , 4) and the Lunelli – Sce hyperoval in PG (2 , 16) are the only hyperovals stabilized by a transitive projective group [10] . In both cases , this group is an irreducible group fixing no triangle in the plane of the hyperoval , nor in a cubic extension of that plane . Using Hartley’s classification ...

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...