Translation planes of order 27

Translation planes of order 27 are classified. Various invariants play an important role in a computer search.     

The translation planes of order 49

In this paper, we describe an exhaustive computer search for all the translation planes of order 49. The search uses the well-known correspondence between translation planes of order 49 and spreads of lines inPG(3,7). We conclude that there are exactly 1347 translation planes of order 49, and give some of their properties including group sizes and 7-ranks.     

Tangentially transitive planes of order 16

Lorimer and Rahilly have given constructions for translation planes of order 16. We show that both of these planes have a subplane of order 2 and a group of collineations fixing pointwise which is transitive on the points of –. We show that all such planes of order 16 are isomorphic and that these planes have a number of other properties which are exceptional.     

On extendable planes of order ten

It is well known that the only way of extending a projective plane of order n (conceivable orders are 2, 4 and 10) is adjoining a set of hyperovals to the given projective plane. A converse is proved in this note. It is shown as a corollary that the existence of an extendable plane of order 10 is equivalent to the existence of a quasi-symmetric 2-(111, 12, 10)-design.     

Collineations of projective planes of order 9

Let P be a projective plane of order 9 other than one of the four known ones. Then the order of the full collineation group of P is 3^a · 5^b, where a is a nonnegative integer and b = 0 or 1.     

Subplanes of projective planes of order 121

We determine orbit representatives of all proper subplanes generated by quadrangles of a Veblen-Wedderburn (VW) plane Π of order 11² and the Hughes plane Σ of order 11² under their full collineation groups. In Π, there are 13 orbits of Baer subplanes all of which are desarguesian and approximately 3000 orbits of Fano subplanes. In Σ, there are 8 orbits of Baer subplanes all of which are desarguesian, 2 orbits of subplanes of order 3 and at most 408,075 distinct Fano subplanes. This work was motivated by the well-known question: “Does there exist a non-desarguesian projective plane of prime order?” The question remains unsettled.     

Subplanes of order 3 in Figueroa planes

In this study we show the existence of subplanes of order 3 in Figueroa planes of odd order.     

Largek-arcs in inversive planes of odd order

An infinite family of largek-arcs in the inversive plane over a finite field GF(q), withq 1 (mod 3),q71 orq {17,23, 27,29,41,47,49,53,59} is constructed.Research supported by G.N.S.A.G.A. of C.N.R., project Applicazioni della matematica per la tecnologia e la società, subproject Calcolo simbolico.     

A new class of square order planes

Only a few classes of square order planes are known. These are generalized André planes (including Hall planes), flag transitive planes, Hering's planes and Walker's planes. A new class of planes of order 5^2r, where r is an odd natural number, is constructed and the translation complements of the corresponding planes have been determined. The translation complements divide the sets of distinguished points into 4 orbits of lengths 1, 1, 5^r ? 1, and 5^2r ? 5^r and it is of order 5^r(5^r ? 1)² if r ≠ 1. In the particular case of r = 1, the translation complement divides the set of distinguished points into two orbits of lengths 6 and 20 and it is of order 480.     

Elations in translation planes of order 16

In this article we prove the following: Let E denote the group generated by the set of all elations in a non-Desarguesian translation plane π of order 16. Then E is elementary abelian or dihedral of order 6 or 10. If ¦E¦=10 then the set of elation axes defines a derivable net contained in an E-invariant Desarguesian net of degree 7 and in this case π is the Hall plane. Thus, the only translation planes of order 16 to admit at least four elations with distinct axes are the Desarguesian and Hall planes.