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.     

Flag transitive planes of orderq n with a long cyclel ∞ as a collineation

Baker and Ebert [1] presented a method for constructing all flag transitive affine planes of orderq ² havingGF(q) in their kernels for any odd prime powerq. Kantor [6; 7; 8] constructed many classes of nondesarguesian flag transitive affine planes of even order, each admitting a collineation, transitively permuting the points at infinity. In this paper, two classes of non-desarguesian flag transitive affine planes of odd order are constructed. One is a class of planes of orderq ⁿ , whereq is an odd prime power andn 3 such thatq ⁿ 1 (mod 4), havingGF(q) in their kernels. The other is a class of planes of orderq ⁿ , whereq is an odd prime power andn 2 such thatq ⁿ 1 (mod 4), havingGF(q) in their kernels. Since each plane of the former class is of odd dimension over its kernel, it is not isomorphic to any plane constructed by Baker and Ebert [1]. The former class contains a flag transitive affine plane of order 27 constructed by Kuppuswamy Rao and Narayana Rao [9]. Any plane of the latter class of orderq ⁿ such thatn 1 (mod 2), is not isomorphic to any plane constructed by Baker ad Ebert [1].The author is grateful to the referee for many helpful comments.     

Polarity and transitive parabolic unitals in translation planes of odd order

A parabolic unital of a translation plane is called transitive, if the collineation group G fixing fixes the point at infinity of and acts transitively on the affine points of . It has been conjectured that if a transitive parabolic unital consists of the absolute points of a unitary polarity in a commutative semi-field plane, then the sharply transitive normal subgroupK of G is not commutative. So far, this has been proved for commutative twisted field planes of odd square order, see [1],[5]. Here we prove this conjecture for commutative Dickson planes. Received 14 May 2001.     

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.     

Two-transitive parabolic ovals

We investigate finite affine planes of even order possessing a parabolic oval and a collineation group G which leaves invariant and acts 2-transitively on its affine points. The main attention is devoted to translation planes. The odd order case has already been considered by Enea and Korchmaros in [5]. Our main result shows that if has even order and possesses two 2-transitive parabolic ovals which share at least two, but not all their affine points, then is Desarguesian. Received 20 July 1998.     

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.     

Die Oktavenebene als Translationsebene mit großer Kollineationsgruppe

It is shown that the affine plane over the Cayley numbers is the only 16-dimensional locally compact topological translation plane having a collineation group of dimension at least 41. This (hitherto unpublished) result is one of the ingredients of H. Salzmann's characterizations of the Cayley plane among general compact projective planes by the size of its collineation group.The proof involves various case studies of the possibilities for the structure and size of collineation groups of 16-dimensional locally compact translation planes. At the same time, these case studies are important steps for a classification program aiming at the explicit determination of all such translation planes having a collineation group of dimension at least 38.     

Derived semifield planes of odd order admitting a dihedral group of affine homologies

Derived semifield planes of odd order admitting non-trivial involutorial affine homologies with more than one axis, are examined in detail, under the assumption that the group generated in the translation complement is dihedral. The whole structure of the semifields S coordinatizing such planes is determined. The class of the semifields S of dimension 4 over their centres is characterized.Dedicated to A. Barlotti on the occasion of his 65. birthday.Research partially supported by G.N.S.A.G.A. (C.N.R.)     

Planar Functions and Planes of Lenz-Barlotti Class II

Planar functions were introduced by Dembowski and Ostrom [4] to describe projective planes possessing a collineation group with particular properties. Several classes of planar functions over a finite field are described, including a class whose associated affine planes are not translation planes or dual translation planes. This resolves in the negative a question posed in [4]. These planar functions define at least one such affine plane of order 3^e for every e 4 and their projective closures are of Lenz-Barlotti type II. All previously known planes of type II are obtained by derivation or lifting. At least when e is odd, the planes described here cannot be obtained in this manner.     

Construction of two-dimensional flag-transitive planes

An affine plane is called flag-transitive if it admits a collineation group which acts transitively on the incident point-line pairs. It has been shown that finite flag-transitive planes are necessarily translation planes, and much work has been devoted to this class of translation planes in recent years. All flag-transitive groups of finite affine planes have been determined, and an infinite family of non-Desarguesian flag-transitive planes has been found. In this paper a method is given for constructing all two-dimensional flag-transitive planes of odd order, subsuming the infinite family mentioned above.     

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.     

The translation planes of order 81 that admit SL(2,5), generated by affine elations

The translation planes of order 81 admitting SL(2, 5), generated by affine elations, are completely determined. There are seven mutually non-isomorphic translation planes, of which five are new. Each of these planes may be derived producing another set of seven mutually non-isomorphic translation planes admitting SL(2, 5), where the 3-elements are Baer. Of this latter set, five planes are new.     

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.     

Translation planes of order ϱ6 admitting SL(2, ϱ2)

Large numbers of translation planes are constructed which have order ?⁶ and admit a collineation group SL(2, ?²) generated by elations.     

Una proprietà gruppale delle involuzioni planari che mutano in sé un'ovale di un piano proiettivo finito

Summary The purpose of the present paper is to prove the following theorem: Let Ω be an oval in the projective plane P of odd order n. If P admits a collineation group G wich maps Ω onto itself and is doubly transitive on Ω, then P is desarguesian, Ω is a conic and G contains all collineations in the little projective group PSL(2, n) of P wich leaves Ω invariant.

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Entrata in Redazione il 5 april 1977.     

A class of non-Desargusian planes

A new class of non-Desargusian planes of order q², where q is a power of an odd prime, is constructed. These planes have the interesting property that they all admit a collineation group of order (q² ? 1).     

Characterization of Translation Planes by Orbit Lengths

The Desarguesian, Hall, and Hering translation planes of order q² are characterized as exactly those translation planes of odd order with spreads in PG (3,q) that admit a linear collineation group with infinite orbits one of length q+1 and i of length (q-q) /i for i=1 or 2.     

Collineation groups of translation planes admitting hyperbolic Buekenhout or parabolic Buekenhout-Metz unitals

It is shown that for every semifield spread in PG(3,q) and for every parabolic Buekenhout-Metz unital, there is a collineation group of the associated translation plane that acts transitively and regularly on the affine points of the parabolic unital. Conversely, any spread admitting such a group is shown to be a semifield spread. For hyperbolic Buekenhout unitals, various collineation groups of translation planes admitting such unitals and the associated planes are determined.     

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.     

SU4 (ℂ) als Kollineationsgruppe in sechzehndimensionalen lokalkompakten Translationsebenen

We study 16-dimensional locally compact translation planes in which, for an affine point o, the stabilizer of the affine collineation group contains a subgroup locally isomorphic to SU₄ (). If has only one affine fixed point o, then it is shown that either the plane is the classical Moufang plane over the Cayley numbers, or else must be normal in the stabilizer and has dimension at most 37. This also comprises the proof of the fact that if contains a subgroup locally isomorphic to SU₄() × SL₂() then the plane is the classical Cayley plane. The case that has more affine fixed points in dealt with as well; then, except for a well-known family of planes admitting Spin₇() as a group of collineations, has dimension at most 34.