SOME CHARACTERIZATIONS OF DESARGUESIAN TRANSLATION PLANES

In this note we consider finite translation planes with large translation complements. In particular, we characterize finite affine Desarguesian translation planes in two ways, according to the existence of subgroups in the translation complement that are divisible by relatively large integers, together with modest additional restrictions.

A flag transitive plane of order49and its translation complement

The translation complement of the flag transitive plane of order49[Proc. Amer. Math. Soc. 32 (1972), 256-262] constructed by Rao is computed. It is shown that the flag transitive group itself is the translation complement and it is a solvable group of order600.

Translation complements of C-planes: (I)

Narayana Rao, Rodabaugh, Wilke and Zemmer constructed a new class of finite translation planes from exceptional near-fields described by Dickson and Zassenhaus. These planes referred to as C-planes are not coordinatized by the generalized André systems. In this paper we compute the translation complement of the C-plane corresponding to the C-system III–1. It is found that the translation complement is of order 6912 and it divides the set of ideal points into two orbits of lengths 2 and 48.

On collineation groups of translation planes of orderq4

LetPbe an affine translation plane of orderq4admitting a nonsolvable groupGin its translation complement. IfGfixes more thanq+1slopes, the structure ofGis determined. In particular, ifGis simple thenqis even andG=L2(2s)for some integersat least2.

On spreads in PG(3,2s) that admit projective groups of order 2s

Let Γ be a spread in = PG(3, q); thus Γ consists of a set of q2 +1 mutually skew lines that partition the points of . Also let Λ be the group of projectivities of that leave Γ invariant: so Λ is the “linear translation complement” of Γ, modulo the kern homologies. Recently, inspired by a theorem of Bartalone [1], a number ofresults have been obtained, in an attempt to describe (Γ, Λ) when q2 divides |Λ|. A good example of such a result is the following theorem of Biliotti and Menichetti [3], which ultimately depends on Ganley's characterization of likeable functions of even characteristic [5].

On a C–plane of order 25

Rao, Rodabaugh, Wilke and Zemmer [J. Combin. Theory Ser. A. 11 (1971), 72–92] constructed a number of new VW systems called C-systems from the exceptional near–fields and established that they coordinatize translation planes not isomorphic to generalized André planes. In this paper the translation complement of the plane coordinatized by the C-system I–1 has been found. This plane has the interesting property that its translation complement divides the ideal points into two orbits of lengths 10 and 16. Further, the translation complement contains a subgroup isomorphic to SL(2,5) and therefore one of the exceptional Walker's planes of order 25 [H. Luneberg, Translation Planes, Springer-Verlag (1980), pp.235–244] is indeed the C–plane corresponding to the C–system I–1, which was discovered in 1969.

A class of translation planes of square order

A class of translation planes of order p2r, where r is an odd natural number and p is a prime, p ≥ 7, p ≢ ± (mod 10) is constructed. A salient feature shared by all these planes is that one ideal point is fixed by the translation complement and the remaining ideal points are divided into at least two orbits, one of which is of length pr.

Translation planes of dimension two and characteristic two

This article discusses translation planes of dimension two and characteristic two. LetGbe a subgroup of the linear translation complement of such a planeπ. The nature ofGand its possible action onπare investigated. This continues previous work of the authors. It is shown that no new groups occur.

A characterization of the desarguesian planes of orderq2bySL(2,q)

The main result is that if the translation complement of a translation plane of orderq2contains a group isomorphic toSL(2,q)and if the subgroups of orderqare elations (shears), then the plane is Desarguesian. This generalizes earlier work of Walker, who assumed that the kernel of the plane containedGF(q).