Two-transitive groups on a hyperbolic unital

A classification given previously of all projective translation planes of order q² that admit a collineation group G admitting a two-transitive orbit of q+1 points is applied to show that the only projective translation planes of order q² admitting a hyperbolic unital acting two-transitively on a secant are the Desarguesian planes and the unital is a Buekenhout hyperbolic unital.     

Elementary abelian 2-groups on the line at infinity of translation planes

Let G be a subgroup of the linear translation complement of a translation plane of order q^d with kernel GF(q) and let ¯G be the factor group modulo the scalars. We show that if ¯G is elementary abelian of order 2^a, and if each involution in ¯G has a conjugate class of length greater than a+1 then 2^e divides d, where e=[1/2(a+1)]–1. We show that one of Walker's planes is a counterexample if we drop the condition on lengths of conjugate classes. The Walker plane in question turns out to be of rank 3. This is one of Walker's planes of order 25 and was not previously known to have rank 3.Dedicated to R. Artzy     

On translation planes of orderq 3 that admit a collineation group of orderq 3

Let II be a translation plane of orderq ³, with kernel GF(q) forq a prime power, that admits a collineation groupG of orderq ³ in the linear translation complement. Moreover, assume thatG fixes a point at infinity, acts transitively on the remaining points at infinity andG/E is an abelian group of orderq ², whereE is the elation group ofG.In this article, we determined all such translation planes. They are (i) elusive planes of type I or II or (ii) desirable planes.Furthermore, we completely determined the translation planes of orderp ³, forp a prime, admitting a collineation groupG of orderp ³ in the translation complement such thatG fixes a point at infinity and acts transitively on the remaining points at infinity. They are (i) semifield planes of orderp ³ or (ii) the Sherk plane of order 27.     

A class of unitals of order q which can be embedded in two different planes of order q2

By deriving the desarguesian plane of order q² for every prime power q a unital of order q is constructed which can be embedded in both the Hall plane and the dual of the Hall plane of order q² which are non-isomorphic projective planes. The representation of translation planes in the fourdimensional projective space of J. André and F. Buekenhouts construction of unitals in these planes are used. It is shown that the full automorphism groups of these unitals are just the collineation groups inherited from the classical unitals.     

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.     

The translation planes of order q2 that admit SL(2,q) as a collineation group. II. Odd order

Let be a translation plane of odd order q², where q=p^r and p is a prime. If admits SL(2,q) (or PSL(2,q)) as a collineation group then is a Desarguesian, Hall, or Hering plane, or one of two Walker planes of order 25.Partially supported by grants from the National Science Foundation.     

Translation planes of large dimension admitting nonsolvable groups

In this article, the question is considered whether there exist finite translation planes with arbitrarily small kernels admitting nonsolvable collineation groups. For any integerN, it is shown that there exist translation planes of dimension >N and orderq ³ admittingGL(2,q) as a collineation group.     

A family of translation planes of order q3

In this note, some new class of translation planes of order q³, where q is an odd prime power with q 3,7, are constructed. The translation complement of any plane of this class has three orbits lengths 1, 1 and q³-1 on 1.     

Spreads,ovoids andS 5

We consider translation planes of orderq ² (whereq andq ² - 1 are coprime to 30) such thatS ₅ acts on the line at infinity. It turns out that the Klein correspondence is in particular useful for the investigation of these planes. Representations of the planes, automorphisms and examples of low order are studied in detail. In view of a problem of Ostrom (Math. Z. 156 (1977), 59–71), series of translation planes are constructed with the following property: the translation complement is nonsolvable and has an order coprime to the characteristic of the plane.     

Baer Involutions in Translation Planes Admitting Large Elation Groups

This article is concerned with compatibility of planar and non-planar involutions in translation planes π of even order q. It is shown that if β is a Baer 2-group and ? is an elation group of order q which normalizes β then ¦β¦ ≤ 2. This extends the analogous theorem of Ganley for semifield planes.     

A translation plane of order 192 admitting SL(2,5), obtained by 12-nest replacement

The problem of determining two-dimensional translation planes admitting SL(2,5) in the translation complement has been intensively studied. Most examples arise from multiple-derivation of a Desarguesian plane. In this paper, we construct two translation planes of order 19² admitting SL(2,5), one of which is obtained by 12-nest replacement.      

A family of translation planes of order q 2m+1 with two orbits of length 2 and q 2m+1−1 on l ∞

A class of translation planes of order q ^2m+1, where q is an odd prime power and m1, is constructed. If m=1, then this class is contained in the class of order q ³ constructed by Hiramine [5]. These planes of order q ^2m+1 are of dimension 2m+1 over their kernels. If q ^2m+13³, then the linear translation complements of these planes have two orbits of length 2 and q ^2m+1–1 on l and this class contains many planes which are not generalized André planes. If q ^2m+1= 3³, then each plane of this class is isomorphic to the Hering plane of order 27.Dedicated to Professor Tuyosi Oyama on his 60th birthday     

Ovoids and translation planes revisited

Kantor has previously described the translation planes which may be obtained by projecting sections of ovoids in ⁺(8, q)-spaces to ovoids in corresponding ⁺(6, q)-spaces. Since the Klein correspondence associates spreads in 4-dimensional vector spaces with ovoids in ⁺(6, q)-spaces, there are corresponding translation planes of order q ² and kernel containing GF(q). In this article, we revisit some of these translation planes and give some presentations of the spreads. Motivated by various properties of the planes, we study, in general, translation planes which admit certain homology groups and/or elation groups. In particular, we develop new constructions of projective planes of Lenz-Barlotti class II-1.Finally, we show how certain projective planes of order q ² of Lenz-Barlotti class II-1 may be considered equivalent to flocks of quadratic cones in PG(3, q).This work was partially supported by NSF grant DMS-8800843.     

More translation planes and semifields from Dembowski–Ostrom polynomials

In this article we use pairs of Dembowski–Ostrom polynomials with special properties (see (P1)–(P3) in the introduction below) to construct translation planes of order q ⁿ which admit cyclic groups of order q ⁿ ?1 having orbits of lengths 1, 1, (q ⁿ ?1)/2, (q ⁿ ?1)/2 on the line at infinity. The same pairs also define semifields of order q ²ⁿ . We discuss the properties of these translation planes and semifields. These constructions extend the related construction in Dempwolff and Müller, Osaka J Math [5].     

Translation Ovoids of Generalized Quadrangles and Hexagnos

We define the notion of a translation ovoid in the classical generalized quadrangles and hexagons of order q, and we enumerate all known examples; translation spreads are defined dually. A modification of the known ovoids in the generalized hexagon H(q), q=3^2h+1, yields new ovoids of that hexagon. Dualizing and projecting along reguli, we obtain an alternative construction of the Roman ovoids due to Thas and Payne. Also, we construct a new translation spread in H(q) for any 1 mod 3, q odd, with the property that any projection along reguli yields the classical ovoid in the generalized quadrangle Q(4,q). Finally, we prove that for q odd, the new example is the only non-Hermitian translation spread in H(q) with the property that any projection along reguli yields the classical ovoid in Q(4,q).     

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.     

Desarguesian partial parallelisms

This article establishes connections between Desarguesian partial parallelisms of deficiency one in PG(3,q) and translation planes of orderq ⁴ admitting a collineation group isomorphic to SL(2,q) which is generated by Baer collineations.The author is grateful to Professor Alan Prince for helpful conversations with respect to this article and, in particular, with respect to the actual bounds for certain maximal partial spreads.The author is grateful to the referee for helpful comments in the writing of this article.     

The autotopism group of ap-primitive semifield plane of orderq 4

Let κ be a semifield plane of order q⁴ with kernel K≅GF(q²), where q=p^r, p is prime. Previously, Johnson determined the form of p-primitive semifield planes of order q⁴, q=p^r, and Cordero specified the form of autotopisms and proved the solvability of an autotopism group for the particular case q=p. The goal of the present article is to give an explication of the form of autotopisms and prove the solvability of an autotopism group in the general case. Translated fromAlgebra i Logika, Vol. 35, No. 3, pp. 334–344, May–June, 1996.     

A characterisation of thas maximal arcs in translation planes of square order

In 1974 J.A. Thas constructed a class of maximal arcs in certain translation planes of order q². We characterise these as being exactly those (non-trivial) maximal arcs that are stabilised by an homology of order q– 1.The first author gratefully acknowledges the support of an Australian Postgraduate Research Award.     

Near nearfield planes

The translation planes of order k that admit at least two affine homology groups of order (k–1)/2 are classified. If k7² or 23², the plane is generalized André and there is a Dickson pair {q, n} such that q ⁿ =k and a cyclic homology subgroup of order (q ⁿ –1)/2n.This article was written when the first author was visiting the University of Iowa during 1990–91.