Collineation Research Articles

Overgroups of Cyclic Sylow Subgroups of Linear Groups

We use a theorem of Guralnick, Penttila, Praeger, and Saxl to classify the subgroups of the general linear group (of a finite dimensional vector space over a finite field) which are overgroups of a cyclic Sylow subgroup. In particular, our results provide the starting point for the classification of transitive m-systems; which include the transitive ovoids and spreads of finite polar spaces. We also use our results to prove a conjecture of Cameron and Liebler on irreducible collineation groups having equally many orbits on points and on lines.

On Rank 2 Complex Reflection Groups

We describe a class of groups with the property that the finite ones among them are precisely the complex reflection groups of rank 2. This situation is reminiscent of Coxeter groups, among which the finite ones are precisely the real reflection groups. We also study braid relations between complex reflections and indicate connections to an axiomatic study of root systems and to the Shephard–Todd “collineation groups.”

Unitals in the regular nearfield planes

Classes of parabolic unitals in the regular nearfield planes of odd square order are enumerated and classified. These unitals correspond to certain Buekenhout-Metz unitals in the classical plane. Their collineation groups are determined and the unitals are sorted by projective equivalence.

Cones for the Moulton planes

We present a two-parameter family of affinely connected surfaces which admit the cylinder group as a collineation group of their geodesics. The Moulton Planes in the radial model of Betten, the circular cone, as well as the real affine plane, are part of this family. The Moulton Planes occur in this family in the same way as the real affine plane is contained in a family of cones with decreasing steepness.

16-dimensional compact projective planes with a collineation group of dimension ≥ 35

A compact projective plane is isomorphic to the classical Moufang plane if it admits an automorphism group of dimension at least 38 which fixes a non-incident point-line pair.

Planes and processes

This article presented to Combinatorics 2006 is a survey of finite projective planes and the processes used to construct them. All non-translation planes are described, fundamental processes in translation planes are defined and some of these are used to connect semi-field flocks with symplectic spreads. Hermitian ovoids are connected to extensions of derivable nets, and three types of ‘lifting’ methods are discussed. Furthermore, hyperbolic fibrations and ‘regulus-inducing’ central collineation groups are connected to flocks of quadratic cones. Finally, hyper-reguli and multiple hyper-regulus replacement are considered.

On the finite projective planes of order up to q4, q odd, admitting PSL(3,q) as a collineation group

On the finite projective planes of order up to q4, q odd, admitting PSL(3,q) as a collineation group

On sharply transitive sets in PG(2,q)

In PG(2, q) a point set K is sharply transitive if the collineation group preserving K has a subgroup acting on K as a sharply transitive permuta- tion group. By a result of Korchmsharply transitive hyperovals only exist for a few values of q, namely q = 2, 4 and 16. In general, sharply transitive complete arcs of even size in PG(2, q) with q even seem to be spo- radic. In this paper, we construct sharply transitive complete 6( p q 1)-arcs for q = 4 2h+1 , h � 4. As far as we are concerned, these are the smallest known complete arcs in PG(2, 4 7 ) and in PG(2, 4 9 ); also, 42 seems to be a new value of the spectrum of the sizes of complete arcs in PG(2, 4 3 ). Our construction applies to any q which is an odd power of 4, but the problem of the completeness of the resulting sharply transitive arc remains open for q � 4 11 . In the second part of this paper, sharply transitive subsets arising as orbits under a Singer subgroup are considered and their characters, that is the possible intersection numbers with lines, are investigated. Subsets of PG(2, q) and certain linear codes are strongly related and the above results from the point of view of coding theory will also be discussed.

Flag-transitive and almost simple orbits in finite projective planes

Let Π be a projective plane of order n and let G be a collineation group of Π with a point-orbit O of length v. We investigate the triple (Π,O,G) when O has the structure of a non trivial 2-(v,k,1) design, G induces a flag-transitive and almost simple automorphism group on O and n≤∑(O)=b+v+r+k.

On the Ree Unital

The embedding of a Ree Unital $${\mathcal{R}}(q)$$ in a finite projective plane ? of order up to q 4 is investigated when the Ree group is induced on $${\mathcal{R}}(q)$$ by a collineation group of ?. In particular, it is shown that such a embedding is not admissible for q ? 3, extending in this way a result of Luneburg dating back to 1965.

The nonexistence of projective planes of order 12 with a collineation group of order 8

AbstractIn this article, we prove that there does not exist a symmetric transversal design ${\rm STD}_2[12;6]$ which admits an automorphism group of order 4 acting semiregularly on the point set and the block set. We use an orbit theorem for symmetric transversal designs to prove our result. As a corollary of the result, we prove that there is no projective plane of order 12 admitting a collineation group of order 8. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 411–430, 2008

Two-transitive groups on a hyperbolic unital

A classification given previously of all projective translation planes of order q 2 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 q 2 admitting a hyperbolic unital acting two-transitively on a secant are the Desarguesian planes and the unital is a Buekenhout hyperbolic unital.

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).

PROPER MATTER COLLINEATIONS OF PLANE SYMMETRIC SPACETIMES

We investigate matter collineations of plane symmetric spacetimes when the energy–momentum tensor is degenerate. There exists three interesting cases where the group of matter collineations is finite-dimensional. The matter collineations in these cases are either four, six or ten in which four are isometries and the rest are proper.

On the nonexistence of certain Hughes generalized quadrangles

In this paper, we prove that the Hermitian quadrangle $${\mathsf{H}}(4, q^2)$$ is the unique generalized quadrangle Γ of order (q 2, q 3) containing some subquadrangle of order (q 2, q) isomorphic to $${\mathsf{H}}(3, q^2)$$ such that every central elation of the subquadrangle is induced by a collineation of Γ.

Complete caps in [formula omitted] with q odd

In PG ( 3 , q ) with q ≡ 3 mod 4 and q ⩾ 7 , a complete ( q 2 + q + 4 ) / 2 -cap is constructed. A characterization in terms of the collineation group preserving such a cap is also provided.

4-Transitivity and 6-figures in some Moufang-Klingenberg planes

In this paper, the Moufang-Klingenberg plane over a local alternative ring R of dual numbers is studied. It is shown that its collineation group is transitive on quadrangles. It is therefore shown that the coordinatization of these Moufang-Klingenberg planes is independent of the choice of the coordinatization quadrangle. Also, the concept of 6-figures is extended to these Moufang-Klingenberg planes and it is shown that any 6-figure corresponds to only one inversible m ∈ R.

Translation planes of order q 2 admitting a two-transitive orbit of length q + 1 on the line at infinity

A classification is given of all translation planes of order q 2 that admit a collineation group G admitting a two-transitive orbit of q + 1 points on the line at infinity.

Irreducible collineation groups with two orbits forming an oval

Let G be a collineation group of a finite projective plane π of odd order fixing an oval Ω. We investigate the case in which G has even order, has two orbits Ω 0 and Ω 1 on Ω, and the action of G on Ω 0 is primitive. We show that if G is irreducible, then π has a G-invariant desarguesian subplane π 0 and Ω 0 is a conic of π 0 .

Projective Planes with a Doubly Transitive Projective Subplane

Projective planes of order up to q 3 with a collineation group G acting 2-transitively on a subplane of order q are investigated.