Collineation Research Articles

Collineation group as a subgroup of the symmetric group

AbstractLet ψ be the projectivization (i.e., the set of one-dimensional vector subspaces) of a vector space of dimension ≥ 3 over a field. Let H be a closed (in the pointwise convergence topology) subgroup of the permutation group $\mathfrak{S}_\psi $ of the set ψ. Suppose that H contains the projective group and an arbitrary self-bijection of ψ transforming a triple of collinear points to a non-collinear triple. It is well known from [Kantor W.M., McDonough T.P., On the maximality of PSL(d+1,q), d ≥ 2, J. London Math. Soc., 1974, 8(3), 426] that if ψ is finite then H contains the alternating subgroup $\mathfrak{A}_\psi $ of $\mathfrak{S}_\psi $. We show in Theorem 3.1 that H = $\mathfrak{S}_\psi $, if ψ is infinite.

Conics in the hyperbolic plane intrinsic to the collineation group

In the manner of Steiner’s interpretation of conics in the projective plane we consider a conic in a planar incidence geometry to be a pair consisting of a point and a collineation that does not fix that point. We say these loci are intrinsic to the collineation group because their construction does not depend on an imbedding into a larger space. Using an inversive model we classify the intrinsic conics in the hyperbolic plane in terms of invariants of the collineations that afford them and provide metric characterizations for each congruence class. By contrast, classifications that catalogue all projective conics intersecting a specified hyperbolic domain necessarily include curves which cannot be afforded by a hyperbolic collineation in the above sense. The metric properties we derive will distinguish the intrinsic classes in relation to these larger projective categories. Our classification emphasizes a natural duality among congruence classes induced by an involution based on complementary angles of parallelism relative to the focal axis of each conic, which we refer to as split inversion (Definition 5.3).

A family of flat Laguerre planes of Kleinewillinghöfer type IV.A

Just like Lenz–Barlotti classes reflect transitivity properties of certain groups of central collineations in projective planes, Kleinewillinghofer types reflect transitivity properties of certain groups of central automorphisms in Laguerre planes. In the case of flat Laguerre planes, Polster and Steinke have shown that some of the conceivable types cannot exist, and they gave models for most of the other types. Only few types are still in doubt. Two of them are types IV.A.1 and IV.A.2, whose existence we prove here. In order to construct our models, we make systematic use of the restrictions imposed by the group generated by all central automorphisms guaranteed in type IV. With these models all simple Kleinewillinghofer types with respect to Laguerre homologies and also with respect to Laguerre homotheties are now accounted for, and the number of open cases of Kleinewillinghofer types (with respect to Laguerre homologies, Laguerre translations and Laguerre homotheties combined) is reduced to two.

OH radical production stimulated by (RW)4D, a synthetic antimicrobial agent and indolicidin

Antimicrobial peptides (AMPs) have been widely studied as a potential class of antibiotics for treating infectious diseases caused by multidrug-resistant (MDR) bacterial strains. While the cell surface and membrane are generally implicated in the antibacterial action of AMPs, we still lack a complete understanding of the detailed mechanism(s) involved. Here we show that the natural antimicrobial peptide indolicidin (ILN) and a more potent synthetic dendrimeric analog, (RW)4D, generate hydroxide radicals in target bacterial cells via a Fenton reaction, as Collins group has reported for bactericidal antibiotics. Our results further suggest that the TCA cycle does not regulate release of irons from iron clusters, and electron donors such as NADH from the TCA cycle are only partially responsible for superoxide generation. Since AMP's are refractory to bacterial resistance, unlike the bacteriocidal antibiotics tested, we hypothesize that the source of the OH radicals differs from that in the latter case.

Cameron–Liebler line classes

New examples of Cameron-Liebler line classes in $\mathrm{PG}(3,q)$ are given with parameter $\frac{1}{2}(q^2 -1)$. These examples have been constructed for many odd values of $q$ using a computer search, by forming a union of line orbits from a cyclic collineation group acting on the space. While there are many equivalent characterizations of these objects, perhaps the most significant is that a set of lines $\mathcal{L}$ in $\mathrm{PG}(3,q)$ is a Cameron-Liebler line class with parameter $x$ if and only if every spread $\mathcal{S}$ of the space shares precisely $x$ lines with $\mathcal{L}$. These objects are related to generalizations of symmetric tactical decompositions of $\mathrm{PG}(3,q)$, as well as to subgroups of $\mathrm{P\Gamma L}(4,q)$ having equally many orbits on points and lines of $\mathrm{PG}(3,q)$. Furthermore, in some cases the line classes we construct are related to two-intersection sets in $\mathrm{AG}(2,q)$. Since there are very few known examples of these sets for $q$ odd, any new results in this direction are of particular interest.

ISOMETRIES OF POLYHEDRAL HILBERT GEOMETRIES

We show that the isometry group of a polyhedral Hilbert geometry coincides with its group of collineations (projectivities) if and only if the polyhedron is not an n-simplex with n ≥ 2. Moreover, we determine the isometry group of the Hilbert geometry on the n-simplex for all n ≥ 2, and find that it has the collineation group as an index-two subgroup. The results confirm several conjectures of P. de la Harpe for the class of polyhedral Hilbert geometries.

Sixteen-dimensional locally compact translation planes with collineation groups of dimension at least 38

Abstract This article completes the classification of all the planes mentioned in the title.

Local Sharply Transitive Actions on Finite Generalized Quadrangles

We classify the finite generalized quadrangles containing a line L such that some group of collineations acts sharply transitively on the ordered pentagons which start with two points of L. This can be seen as a generalization of a result of Thas and the second author [22] classifying all finite generalized quadrangles admitting a collineation group that acts transitively on all ordered pentagons, although the restriction to sharp transitivity is essential in our arguments. However, the conclusion is exactly the same family of classical generalized quadrangles (the orthogonal quadrangles and their duals). Our main result thus provides a local group theoretic characterization of these classical quadrangles.

Relation of the Mandibular Body and Ramus in Treacher Collins Syndrome

Treacher Collins syndrome is associated with a clockwise mandibular deformity (Plast Reconstr Surg 2008;121:77e-84e). We hypothesize that the Treacher Collins mandible exhibits a more obtuse relation between the body and the ramus compared with unaffected subjects. The purpose of this study was to objectively analyze the body-ramus relation in Treacher Collins compared with control subjects and to investigate the effect of age. This was a retrospective analysis involving three-dimensional computed tomography (CT) scans. Demographic and CT scan information was obtained for Treacher Collins and control subjects. Measurements were performed on the IDXrad platform. The ramus body angle (RBA) was defined as the intersection of the anterior ramus with the tangent from the alveolar process of the mandible. The mandibular plane angle (MPA) was measured in standard fashion. Left and right hemimandibles were analyzed. Statistical analysis involved the paired t-test (P < 0.005 significance). Subgroups based on age, younger than 3 years and older than 12 years, were then compared, within and between groups. Three-dimensional CT scans were identified in 18 Treacher Collins (36 sides) and 24 control (48 sides) subjects. The MPA was excluded in 6 Treacher Collins and 2 control hemimandibles for absent centric occlusion. The Treacher Collins group consisted of 7 females and 11 males, ranging in age from 0.66 to 204 months (mean, 91.6 months). The control cohort included 13 females and 11 males, aged 0.26 to 204 months (mean, 111.5 months). The mean RBAs for the Treacher Collins and comparison groups were 150.28 and 114.2 degrees (P < 0.0001). The mean MPAs were 38.25 and 31.9 degrees for the Treacher Collins group and control subjects, respectively (P < 0.0001). In children younger than 3 years, the RBAs were 154.40 and 119.36 degrees, and for patients older than 12 years, the RBAs were 148.40 and 111.75 degrees for the Treacher Collins group and control subjects (both P < 0.0001). Comparing Treacher Collins younger than 3 years with those older than 12 years, the mean RBAs were not statistically different at 154.40 and 148.40 degrees (P = 0.1869). Among control subjects younger than 3 years compared with older than 12 years, the RBAs approached significance, but not statistically at 119.36 and 111.75 degrees (P = 0.0122). Notably, among control subjects, the MPA did statistically flatten with increasing age, showing 37.36 and 29.50 degrees (P = 0.0011). The clockwise relation of the mandibular body relative to the ramus is objectively documented in Treacher Collins subjects. This relation persists despite increasing age, in distinction to control subjects where the mandibular plane flattens in adolescence.

Transitive groups on the line at infinity of a finite affine plane

The general structure of collineation groups with a faithful transitive action on the line at infinity of a finite affine plane is determined. Moreover, the projective planes admitting a collineation group which fixes a point and acts transitively on a blocking set are classified when the group is of even order.

Subplanes of projective planes of order 121

We determine orbit representatives of all proper subplanes generated by quadrangles of a Veblen-Wedderburn (VW) plane Π of order 112 and the Hughes plane Σ of order 112 under their full collineation groups. In Π, there are 13 orbits of Baer subplanes all of which are desarguesian and approximately 3000 orbits of Fano subplanes. In Σ, there are 8 orbits of Baer subplanes all of which are desarguesian, 2 orbits of subplanes of order 3 and at most 408,075 distinct Fano subplanes. This work was motivated by the well-known question: “Does there exist a non-desarguesian projective plane of prime order?” The question remains unsettled.

Affine planes admitting a collineation group with a transitive action on the line at infinity

The affine planes admitting a collineation group with a transitive action on the line at infinity are investigated and an essentially complete classification is achieved when the group is almost simple. A rather deep result is also obtained when the action on the line at infinity is faithful and primitive.

Collineation groups with one or two orbits on the set of points not on an oval and its nucleus

Projective planes of even order admitting a collineation group fixing an oval and having one or two orbits on the set of points not on the oval and its nucleus are investigated.

The group fixing a completely regular line–oval

We prove that the action of the full collineation group of a symplectic translation plane of even order on the set of completely regular line---ovals is transitive. This provides us with a complete description of the group of collineations fixing a completely regular line---oval.

Transitive bislim geometries of gonality 3, Part II: The group theoretic cases

We consider point–line geometries having three points on every line, having three lines through every point ( bislim geometries), and containing triangles. We classify such geometries under the hypothesis of the existence of a collineation group acting transitively on the point set.

A classification of transitive ovoids, spreads, and m-systems of polar spaces

Many of the known ovoids and spreads of finite polar spaces admit a transitive group of collineations, and in 1988, P. Kleidman classified the ovoids admitting a 2-transitive group. A. Gunawardena has recently extended this classification by determining the ovoids of the seven-dimensional hyperbolic quadric which admit a primitive group. In this paper we classify the ovoids and spreads of finite polar spaces which are stabilised by an insoluble transitive group of collineations, as a corollary of a more general classification of m-systems admitting such groups.

A new family of locally compact 4-dimensional translation planes admitting a 7-dimensional collineation group

Abstract The present paper contains the construction of a family of 4-dimensional translation planes with a 7-dimensional collineation group. The planes of this family do not appear in the comprehensive classification [Betten, J. Reine Angew. Math. 285: 126–148, 1976, Betten, Geometriae Dedicata 3: 405–440, 1974/75, Betten, Math. Z. 154: 125–141, 1977]. We furthermore determine the isomorphism types of the new planes.

Classification of Projective Translation Planes of Order q2 Admitting a Two-Transitive Orbit of Length q + 1

A classification is given of all projective translation planes of order q2 that admit a collineation group G admitting a two-transitive orbit of q + 1 points.

Finite projective planes of order n with a 2 -transitive orbit of length n/2 - 1

The finite projective planes \({\prod}\) of order n with a collineation group G acting 2- transitively on a point subset \({{\mathcal{O}}}\) of \({\prod}\) of size n/2 - 1 are essentially classified.

On affine difference sets and their multipliers

Let D be an affine difference set of order n in an abelian group G relative to a subgroup N . We denote by π ( s ) the set of primes dividing an integer s ( > 0 ) and set H ∗ = H ∖ { ω } , where H = G / N and ω = ∏ σ ∈ H σ . In this article, using D we define a map g from H to N satisfying for τ , ρ ∈ H ∗ , g ( τ ) = g ( ρ ) iff { τ , τ − 1 } = { ρ , ρ − 1 } and show that ord o ( σ ) ( m ) / ord o ( g ( σ ) ) ( m ) ∈ { 1 , 2 } for any σ ∈ H ∗ and any integer m > 0 with π ( m ) ⊂ π ( n ) . This result is a generalization of J.C. Galati’s theorem on even order n [J.C. Galati, A group extensions approach to affine relative difference sets of even order, Discrete Mathematics 306 (2006) 42–51] and gives a new proof of a result of Arasu–Pott on the order of a multiplier modulo exp ( H ) ([K.T. Arasu, A. Pott, On quasi-regular collineation groups of projective planes, Designs Codes and Cryptography 1 (1991) 83–92] Section 5).