A translation plane of order 112

A translation plane of order 11² is constructed. Its translation complement is a solvable group of order 1200 and has 9 orbits on the line at infinity. These orbits have lengths 30, 20, 20, 12, 10, 10, 10, 5, 5, respectively.     

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.      

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     

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.     

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 structure theory for two-dimensional translation planes of order q 2 that admit collineation groups of order q 2

This paper is devoted to the study of translation planes of order q ² and kernel GF(q) that admit a collineation group of order q ² in the linear translation complement. We give a representation of this group by a suitable set of matrices depending on some functions over GF(q). Using this representation we obtain several results concerning the existence and the collineation group of the plane.     

Collineation groups irreducible on the components of a translation plane

We investigate finite translation planes of odd dimension over their kernels in which the translation complement induces on each component l a permutation group whose order is divisible by a p-primitive divisor. Using results of this investigation, we show that rank 3 affine planes of odd dimension over their kernels are either generalized André planes or semi-field planes. A similar result is given for translation planes having a collineation group which is doubly transitive on each affine line; besides the above two possibilities, there is a third possibility; the plane has order 27, the translation complement is doubly transitive on , and SL(2, 13) is contained in the translation complement.We also consider translation planes of odd dimension over their kernels which have a collineation group isomorphic to SL(2, w) with w prime to 5 and the characteristic, and having no affine perspectivity. We show that such planes have order 27, the prime power w=13, and the given group together with the translations forms a doubly transitive collineation group on {ie153-1}. This indicates quite strongly that the Hering translation plane of order 27 is unique with respect to the above properties.Both authors supported in part by NSF Grant No. MCS76-0661 A01.     

A translation plane of order 25

In this paper a new translation plane of order 25 is constructed. It has a collineation group acting on the line at infinity as a permutation group Z of order 48 with the properties:

1. Z contains a normal subgroup 1/2M of order 3 such that Z/1/2M is the direct product of an involution with a dihedral group of order 8.
2. The orbits of Z have lengths 2, 12, 12.

     

On translation planes of orderq 2 that admit a group of orderq 2 (q−1); bartolone's theorem

This article is concerned with translation planesP of orderq ² and kernelK isomorphic toG F(q). IfP admits a collineation groupG in the linear translation. complement and the order ofG K/K isq ²(q?1) then it is shown thatP is either a semifield plane or is a Lüneburg-Tits, Walker or Betten plane. This generalizes earlier work of Bartolone.     

Simple groups acting on translation planes

We discuss the possibility of finite simple groups acting as collineation groups on finite translation planes of odd order with special attention paid to the sporadic simple groups. We assume such a group acts irreducibly (in the vector space sense) on the plane. It is shown that if the characteristic of the plane does not divide the order of the group, then the group cannot be one of eleven sporadic simple groups. Also, if one of the Mathieu groups acts irreducibly on a finite translation plane then it is either M₁₁ or M₂₃.     

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

Let II be a translation plane of orderq ³ with kernel GF(q) that admits a collineation groupG of orderq ³ in the linear translation complement such thatG fixes a point at infinity and acts transitively on the remaining points at infinity.In this paper, we show that any such translation plane II is one of the following types of planes:     

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.     

A characterization of Hering's plane of order 27

Hering's translation plane of order 27 has been characterized by its order and the fact that it admits SL(2, 13) in its translation complement (see [1]). We show that, aside from the Desarguian plane and a Generalized, André plane, it is the only plane of order 27 which admits a subgroup of SL(2, 13) of order 13×12.Partially supported by FONDECYT 0343 and ANDES FOUNDATION.     

The uniqueness of Hering's translation plane of order 27

It is shown that the following conjecture of Kallaher and Ostrom [2] is correct: Hering's translation plane of order 27 is the only translation plane of odd dimension over its kernel which has a collineation group isomorphic to SL(2, w) with w prime to 5 and to the characteristic, and having no affine perspectivity.     

A note on finite semifield planes that admit affine homologies

Let be a translation plane of order n that admits an abelian group of order n in its translation complement. If admits an affine homology then it is a semifield plane.     

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.     

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.     

Transitive affine planes admitting elations

Affine planes which admit a point transitive collineation group and at least one affine elation are considered. Such a plane is shown to be (A,?_∞)-transitive for some point A on ?_t8 and to be a translation plane if at least two distinct elation centers exist. If the plane has at least (order)^1/2+1 distinct elation centers and the group generated by the elations is nonsolvable then the plane is either Desarguesian or Lüneburg-Tits.     

Note on the existence of translation nets

We exhibit a class of (2 ^k + 1, 2^2k )-translation nets with nonabelian translation group, for any natural k. At the same time, it is the first infinite class of translation nets known to admit nonisomorphic translation groups for each of its elements.