Shear planes

A shear plane is a 2n-dimensional stable plane admitting a quasi-perspective collineation group which is a vector group of the same dimension 2n and fixes no point. We show that all of these planes can be derived from a special kind of partial spreads by a construction analogous to the construction of (punctured) dual translation planes from compact spreads. Finally we give a criterion (and examples) for shear planes which are not isomorphic to an open subplane of a topological projective plane.     

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.     

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.     

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.     

Some inherited maximal arcs in derived dual translation planes

In 1974 J. A. Thas constructed a class of maximal arcs in certain translation planes of orderq ². In this paper a new class of maximal arcs is constructed in certain derived dual translation planes that are inherited from the duals of the Thas maximal arcs. It is noted that some (but not all) of the maximal arcs are isomorphic to a class constructed by the author.The author gratefully acknowledges the support of an Australian Postgraduate Research Award.     

Semifield planes of order p 4 and kernel GF(p 2)

In this article we determine the number of non-isomorphic semifield planes of order p⁴ and kernel GF(p²) for p prime, 3 ≤ p ≤ 11. We show that for each of these values of p, the plane is either desarguesian, p-primitive, or a generalized twisted field plane. We also show that the class of p-primitive planes is the largest. We also discuss the autotopism group of the semifields under study.     

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.      

The classification of spreads in PG(3, q) admitting linear groups of order q(q+1), I. odd order

A classification is given of all spreads in PG(3, q), q = p^r, p odd, whose associated translation planes admit linear collineation groups of order q(q +1) such that a Sylow p-subgroup fixes a line and acts non-trivially on it.The authors are indebted to T. Penttila for pointing out the special examples of conical flock translation planes of order q² that admit groups of order q(q+1), when q = 23 or 47.     

A combinatorial theory of Grünbaum's new regular polyhedra,Part II: Complete enumeration

The new regular polyhedra as defined by Branko Grünbaum in 1977 (cf. [5]) are completely enumerated. By means of a theorem of Bieberbach, concerning the existence of invariant affine subspaces for discrete affine isometry groups (cf. [3], [2] or [1]) the standard crystallographic restrictions are established for the isometry groups of the non finite (Grünbaum-)polyhedra. Then, using an appropriate classification scheme which—compared with the similar, geometrically motivated scheme, used originally by Grünbaum—is suggested rather by the group theoretical investigations in [4], it turns out that the list of examples given in [5] is essentially complete except for one additional polyhedron.So altogether—up to similarity—there are two classes of planar polyhedra, each consisting of 3 individuals and each class consisting of the Petrie duals of the other class, and there are ten classes of non planar polyhedra: two mutually Petrie dual classes of finite polyhedra, each consisting of 9 individuals, two mutually Petrie dual classes of infinite polyhedra which are contained between two parallel planes with each of those two classes consisting of three one-parameter families of polyhedra, two further mutually Petrie dual classes each of which consists of three one parameter families of polyhedra whose convex span is the whole 3-space, two further mutually Petrie dual classes consisting of three individuals each of which spanE ³ and two further classes which are closed with respect to Petrie duality, each containing 3 individuals, all spanningE ³, two of which are Petrie dual to each other, the remaining one being Petrie dual to itself.In addition, a new classification scheme for regular polygons inE ⁿ is worked out in §9.     

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 characterisation of spreads ovally-derived from Desarguesian spreads

We characterise all spreads that are obtainable from Desarguesian spreads by replacing a partial spread consisting of translation ovals; the corresponding ovally-derived planes are generalised André planes, of order 2 ^N , although not all generalised André planes are ovallyderived from Desarguesian planes. Our analysis allows us to obtain a complete classification of all nearfield planes that are ovally-derived from Desarguesian planes. It turns out that whether or not a nearfield plane is ovally-derived from a Desarguesian plane depends solely on the parametersq andr, where GF (q) is the kern, andr is the dimension of the plane. Our results also imply that a Hall plane of even orderq ² can be ovally-derived from a Desarguesian spread if and only ifq is a square.     

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.     

Spreads and ovoids in finite generalized quadrangles

This paper is a survey on the existence and non-existence of ovoids and spreads in the known finite generalized quadrangles. It also contains the following new results. We prove that translation generalized quadrangles of order (s,s ²), satisfying certain properties, have a spread. This applies to three known infinite classes of translation generalized quadrangles. Further a new class of ovoids in the classical generalized quadranglesQ(4, 3 ^e ),e3, is constructed. Then, by the duality betweenQ(4, 3 ^e ) and the classical generalized quadrangleW (3 ^e ), we get line spreads of PG(3, 3 ^e ) and hence translation planes of order 3^2e . These planes appear to be new. Note also that only a few classes of ovoids ofQ(4,q) are known. Next we prove that each generalized quadrangle of order (q ²,q) arising from a flock of a quadratic cone has an ovoid. Finally, we give the following characterization of the classical generalized quadranglesQ(5,q): IfS is a generalized quadrangle of order (q,q ²),q even, having a subquadrangleS isomorphic toQ(4,q) and if inS each ovoid consisting of all points collinear with a given pointx ofS\S is an elliptic quadric, thenS is isomorphic toQ(5,q).     

Maximal Baer groups in translation planes and compatibility with homology groups

The translation planes with spreads in PG(3, q) that admit at least two Baer groups of order q–1 are classified.     

A characterization of quaternion planes

The eight-dimensional planes admitting SL₂ as a group of automorphisms are determined.Dedicated to my teacher, Prof. H. Salzmann, on his 60th birthday     

Smooth Stable Planes and the Moduli Spaces of Locally Compact Translation Planes

 Smooth stable planes have been introduced in [3]. At every point p of a smooth stable plane the tangent spaces of the lines through p form a compact spread (see the definition in Section 2) on the tangent space thus defining a locally compact topological affine translation plane . We introduce the moduli space of isomorphism classes of compact spreads, . We show that for the topology of is not by constructing a sequence of non-classical spreads in that converges to the classical spread in , where . Moreover, we prove that the isomorphism type of varies continuously with the point p. Finally, we give examples of smooth affine planes which have both classical and non-classical tangent translation planes. (Received 15 April 1999; in revised form 22 October 1999)     

Transitive Partial Parallelisms of Deficiency One

A new construction of parallelisms, determined by Johnson, is valid for both the finite and infinite cases and gives a variety of partial parallelisms of deficiency one that admit a transitive group. Since there are extensions to parallelisms, one obtains parallelisms admitting a collineation group fixing one spread and transitive on the remaining spreads. The construction permits a counting of the isomorphism classes of the parallelisms. In this article, we enumerate the isomorphism classes of the parallelisms and show that there are at least 1  +  [(q −  3) / 2 r ] mutually non-isomorphic parallelisms in PG(3,q  =  p^r), for p odd. Furthermore, we provide a group-theoretic characterization of the constructed parallelisms.     

Strongly inequivalent representations and Tutte polynomials of matroids

We develop constructive techniques to show that non-isomorphic 3-connected matroids that are representable over a fixed finite field and that have the same Tutte polynomial abound. In particular, for most prime powers q, we construct infinite families of sets of 3-connected matroids for which the matroids in a given set are non-isomorphic, are representable over GF(q), and have the same Tutte polynomial. Furthermore, the cardinalities of the sets of matroids in a given family grow exponentially as a function of rank, and there are many such families.In Memory of Gian-Carlo Rota     

Elementary constructions of locally compact affine planes

In this paper we describe several elementary constructions of 4-, 8- and 16-dimensional locally compact affine planes. The new planes share many properties with the classical ones and are very easy to handle. Among the new planes we find translation planes, planes that are constructed by gluing together two halves of different translation planes, 4-dimensional shift planes, etc. We discuss various applications of our constructions, e.g. the construction of 8- and 16-dimensional affine planes with a point-transitive collineation group which are neither translation planes nor dual translation planes, the proof that a 2-dimensional affine plane that can be coordinatized by a linear ternary field with continuous ternary operation can be embedded in 4-, 8- and 16-dimensional planes, the construction of 4-dimensional non-classical planes that admit at the same time orthogonal and non-orthogonal polarities. We also consider which of our planes have tangent translation planes in all their points. In a final section we generalize the Knarr-Weigand criterion for topological ternary fields.This research was supported by a Feodor Lynen fellowship.