Hughes Planes Research Articles

On PGL(2,q)-invariant unitals embedded in Desarguesian or in Hughes planes

In this paper it is shown that any unital embedded either in PG(2,q2) or in the Hughes plane of order q2, which is invariant under PGL(2,q)×〈σ〉, where σ is the Frobenius involution, is Hermitian or classical Rosati, respectively.

Subplanes of Order $3$ in Hughes Planes

In this study we show the existence of subplanes of order $3$ in Hughes planes of order $q^2$, where $q$ is a prime power and $q \equiv 5 \ (mod \ 6)$. We further show that there exist finite partial linear spaces which cannot embed in any Hughes plane.

A nearfield-free definition of regular Hughes planes and some embeddings of PG(3,q) in Hughes planes of order q 4

We give a nearfield-free definition of some finite and infinite incidence systems by means of half-points and half-lines and show that they are projective planes. We determine a planar ternary ring for these planes and use it to determine the full collineation group and to demonstrate some embeddings of these planes among themselves. We show that these planes include all finite regular Hughes planes and many infinite ones. We also show that PG(3, q) embeds in Hu(q 4) (and show infinite versions of this embedding).

Four-dimensional compact projective planes admitting an affine Hughes group

By constructing all 4-dimensional affine Hughes planes that are not translation planes, we contribute the last missing part to the classification of all compact 4-dimensional projective planes having a non-solvable automorphism group of dimension at least 6. We use the information on the possible actions of the automorphism group obtained by the third author [5] and the construction principle employing derived planes that was given by the second author [4]. The unique solvabilty of certain nonlinear systems of equations has to be shown; this is done using topological degree techniques and delicate analytic arguments. Also the proof of continuity of the geometric operations is quite subtle.

Smooth Hughes planes are classical

We prove that the only compact projective Hughes planes which are smooth projective planes are the classical planes over the complex numbers $\Bbb C $ , the quaternions $\Bbb H $ , and the Caley numbers $\Bbb O $ . As a by-product this shows that an 8-dimensional smooth projective plane which admits a collineation group of dimension $d \geq 17$ is isomorphic to the quaternion projective plane ${\cal P _2\Bbb H }$ . For topological compact projective planes this is true if $d \geq 19$ , and this bound is sharp.

Characterization of 16-dimensional Hughes planes

The well-known finite Hughes planes have compact analoga with 16-dimensional point space. The automorphism group of such a plane is a 36-dimensional Lie group. Theorem: Assume that the compact projective plane $\cal P $ is not isomorphic to the classical Moufang plane over the octonions. Let $\Delta $ be a closed subgroup of $\hbox {Aut} \,\cal P $. If $\dim \Delta \ge 31$ and if $\Delta $ has a normal torus subgroup, then $\cal P $ is a Hughes plane, $\Delta = \hbox {Aut} \,\cal P $, and $\Delta ^{\prime } \cong \hbox {PSL}_3 \Bbb H $.

Affine Hughes groups acting on 4-dimensional compact projective planes

Suppose that the real projective plane is contained as a Baer subplane in a compact 4-dimensional plane . Salzmann [15] has shown that if the group of projective collineations of extends to , then is isomorphic to the complex plane. This means that there are no compact 4-dimensional Hughes planes. If the group AGL2ℝ of offine collineations of extends to , then we call the extended group an offine Hughes group. Here, non-classical examples exist. We show that, up to duality, every action of AGL2ℝ on a compact 4-dimensional plane is an offine Hughes group, and we describe explicitly the possible actions on the point space of as transformation groups. In subsequent work (jointly with N. Knarr and H. Klein), this will be used to determine the possible planes , as well. This will complete the classification of 4-dimensional planes admitting a non-solvable group of dimension at least 6.

A New Class of Unitals in the Hughes Plane

A new class of unitals in the Hughes planes is enumerated and classified. The unital obtained by L. A. Rosati is shown to be a member of this class. Their collineation groups are determined and the unitals are sorted by projective equivalence. The dual designs are described and certain members are shown to be self-dual.

Pseudo-Homogeneous Coordinates for Hughes Planes

AbstractAmong the projective planes, the class of Hughes planes has received much interest, for several good reasons. However, the existing descriptions of these planes are somewhat unsatisfactory. We introduce pseudo-homogeneous coordinates which at the same time are easy to handle and give insight into the action of the group that is generated by all elations of the desarguesian Baer subplane of a Hughes plane. The information about the orbit decomposition is then used to give a description in terms of coset spaces of this group. Finally, we exhibit a non-closing Desargues configuration in terms of coordinates.

Actions of almost simple groups on compact eight-dimensional projective planes are classical, almost

We show that the existence of an almost simple group of automorphisms of dimension greater than 10 characterizes the Hughes planes (including the quarternion plane) among the 8-dimensional compact projective planes.

Locally Compact Hughes Planes

AbstractAmong the eight-dimensional stable planes, the compact connected generalized Hughes planes and the geometries induced on the outer points are characterized by the property that these planes admit an effective action of the group SL3 ℂ.

Partitions of Desarguesian and Hughes planes into complete and incomplete arcs

The purpose of the present article is to obtain partitions of Desarguesian projective planes PG (2,q2), and of Hughes planes, too, into a number of q2+q+1 equicardinal arcs. In a previous paper we showed that PG(2,q2) is a disjoint union of arcs as above, which, as it turned out later, were also complete. We now prove that a similar partition can be obtained for the Hughes planes. We also partition Desarguesian and Hughes planes into incomplete arcs with the same number of points.

Characterizations of the Generalized Hughes Planes

Let be a projective plane and a subplane of . If l is a line of , we let denote the group of all elations in that have as axis and leave Q invariant. In [12, p. 921], Ostrom asked for a description of all finite planes that have a Baer subplane with the property that for all lines l of . Here denotes the order of G. Both the desarguesian planes of square order and the generalized Hughes planes have this property (Hughes [10], Ostrom [14], Dembowski [6]). One of the aims of this paper is to show that these are the only planes having such a Baer subplane.

The combinatorial structure of the Hughes–Zassenhaus plane of order 25

AbstractThe Hughes–Zassenhaus plane, , of order 25 is the simplest of the exceptional Hughes planes (see, e.g. (1), p. 391).In this paper, an incidence table is constructed analogous to tables for the field plane and the regular Hughes plane, and the following rather surprising properties of the plane emerge:(i) The table has much greater internal symmetry than either of the analogous tables,(ii) any polarity with regard to a conic in the central subplane Δ0 of extends to two pairs of polarities in one pair with 30 and the other with 60 singular points each in addition to the six in Δ0.

Generalized Hughes Planes

The projective planes discovered in 1957 by Hughes [3] were originally described by means of a nearfield F satisfying the following conditions:(a) F is finite,(b) the centre and kernel of F coincide,(c) F is of rank 2 over its kernel.(The definitions of these terms will be given in § 2; the terminology used throughout the paper is that of [1].)Rosati [5] showed in 1960 that condition (a) is not necessary, thus constructing the first “infinite Hughes planes”. Condition (b), however, plays an essential part also in Rosati's work.The aim in this paper is to show that condition (b) is superfluous as well. For the finite case, this has been remarked by Ostrom [4] without proof; here we shall show that a “generalized Hughes plane” can be constructed over any nearfield satisfying condition (c) only.

A Characterization of the Hughes Planes

Baer (1) introduced the term "(p,L)-collineation" to denote a central collineation with centre p and axis L. We shall find it convenient to use a modification of the related notion of "(p, L)-transitivity."Definition. Let π0 be a subplane of the projective plane π. Let L be a fixed line of π0, and let p be a fixed point of π0. Let r and s be any two points of π0 that are collinear with p, distinct from p, and not on L. If, for each such choice of r and s, there is a (p, L)-collineation of π that (1) carries π0 into itself and (2) carries r into s, we shall say that π is (p, L, π0)-transitive.