Collineation Research Articles

Collineation groups of octonionic and split-octonionic planes

We present a Veronese formulation of the octonionic and split-octonionic projective and hyperbolic planes. This formulation of the incidence planes highlights the relationship between the Veronese vectors and the rank-1 elements of the Albert algebras over octonions and split-octonions, yielding to a clear formulation of the relationship with the real forms of the Lie groups arising as collineation groups of these planes. The Veronesean representation also provides a novel and minimal construction of the same octonionic and split-octonionic planes, by exploiting two symmetric composition algebras: the Okubo algebra and the paraoctonionic algebra. Besides the intrinsic mathematical relevance of this construction of the real forms of the Cayley–Moufang plane, we expect this approach to have implications in all mathematical physics related with exceptional Lie Groups of type [Formula: see text] and [Formula: see text].

An essay on Freudenthal-Tits polar spaces

We study some properties of the nonembeddable polar spaces related to octonion division rings (and related to the index E7,328). More exactly, we classify its subspaces and show that its generating rank is equal to 5, whereas it was always believed to be at least 6. We also study self-projectivities of length 3 in the maximal singular subspaces, and these turn out to be polarities of octonion projective planes. We establish a connection between the conjugacy classes of such polarities and the orbits of the collineation group of the polar space on triples of opposite singular planes. Along the way, we classify all polar spaces for which each self-projectivity of length 3 in a maximal singular subspace is a polarity.

Projective planes of order 12 do not have a collineation group of order 4

AbstractIn this paper, we prove that there are no projective planes of order 12 admitting a collineation group of order 4. This yields that the order of any collineation group of a projective plane of order 12 is 1, 2, or 3.

Programs used in the paper "Projective planes of order 12 do not have a collineation group of order 4"

Programs used in the paper "Projective planes of order 12 do not have a collineation group of order 4"

A geometrical interpretation of Okubo Spin group

In this work we define, for the first time, the affine and projective plane over the real Okubo algebra, showing a concrete geometrical interpretation of its Spin group. Okubo algebra is a flexible, composition algebra which is also a not unital division algebra. Even though, Okubo algebra has been known for more than 40 years, we believe that this is the first time the algebra was used for affine and projective geometry. After showing that all axioms of affine geometry are verified, we define a projective plane over Okubo algebra as completion of the affine plane and directly through the use of Veronese coordinates. We then present a bijection between the two constructions. Finally we show a geometric interpretation of Spin(O) as the group of collineations that preserve the axis of the plane.

2-элементы в группе автотопизмов полуполевой проективной плоскости

We investigate the well-known hypothesis of D.R. Hughes that the full collineation group of non-Desarguesian semifield projective plane of a finite order is solvable (the question 11.76 in Kourovka notebook was written down by N.D. Podufalov). The spread set method is used to construct the semifield projective planes with cyclic 2- subgroup of autotopisms in the case of linear space of any dimension over the field of prime order. This study completes the analogous considerations of elementary abelian 2-subgroups. We obtain the natural restriction to the order of 2-element for the semifield planes for odd and even order. It is proved that some projective linear groups can not be the autotopism subgroups for the infinite series of semifield planes. The matrix representation of Baer involution allows us to define the geometric property of autotopism of order 4. We can choose the base of a linear space such that the matrix representation of these autotopisms is convenient and uniform, it does not depend on the space dimension. The minimal counter-example is constructed to explain the restriction to the plane order. As a corollary, we proved the solvability of the full collineation group when the nonDesarguesian semifield plane has a certain even order and all its Baer subplanes are also non-Desarguesain. The main results can be used as technical for the further studies of the subgroups of even order in an autotopism group for a finite non-Desarguesian semifield plane. The results obtained are useful to investigate the semifield planes with the autotopism subgroups from J.G. Thompson’s list of minimal simple groups.

Cameron-Liebler line classes with parameter [formula omitted

Cameron-Liebler line classes were introduced in [5], and motivated by a question about orbits of collineation groups of PG(3,q). These line classes have appeared in different contexts under disguised names such as Boolean degree one functions, regular codes of covering radius one, and tight sets. In this paper we construct an infinite family of Cameron-Liebler line classes in PG(3,q) with new parameter x=(q+1)2/3 for all prime powers q congruent to 2 modulo 3. The examples obtained when q is an odd power of two represent the first infinite family of Cameron-Liebler line classes in PG(3,q), q even.

4-Transitivity and Cross-Ratio in Moufang-Klingenberg Planes over Rings of Plural Numbers

In this paper, we carry the obtained results for a certain class of Moufang-Klingenberg planes co-ordinatized by the local alternative ring of dual numbers, over to a class of Moufang-Klingenberg planes coordinatized by the plural algebra of order m which is a local alternative ring. So, we show that its collineation group is transitive on quadrangles and therefore that coordinatization of the planes is independent of the choice of the coordinatization quadrangle. Moreover, by a conjugacy class we discuss the definition of cross-ratio of four distinct points on the projective lines for the planes.

Sixteen-dimensional locally compact planes of Lenz-type V on which {mathrm{SU}}_2 {mathbb {H}} acts as a group of collineations

We explicitly construct all the planes mentioned in the title.

On a new class of Moufang–Klingenberg planes

In this paper, we deal with a new class of Moufang–Klingenberg planes and study with the concepts of 4-transitivity and 6-figures on the class of planes. So, we reach some algebraic and geometric results related on the concepts. For instance, we show that the group of collineations of these planes acts transitively on 4-gons. Finally, we obtain some properties of 6-figures.

Finite collineation groups and birational rigidity

We classify finite groups G in $${\mathrm {PGL}}_{4}({\mathbb {C}})$$ such that $${\mathbb {P}}^3$$ is G-birationally rigid.

A new characterization of the desarguesian and the Figueroa plane

A result of Dempwolff [4] asserts that a projective plane Π of order q3 admitting G≅PGL(3,q) as a collineation group contains a G-invariant subplane π0 isomorphic to PG(2,q) on which G acts 2-transitively. Moreover, G splits the point set and the line set of Π into 3 orbits Pi(Π) and Li(Π), i=1,2,3, consisting of the points (respectively lines) of Π incident with q+1, 1 or 0 lines (respectively points) of π0. In this paper it is proven that Π is either the desarguesian or the Figueroa plane of order q3 if, and only if, (P3(Π),L3(Π)) is isomorphic to (P3(D),L3(D)), where D≅PG(2,q3).

On some 3-primitive projective planes

We evolve an approach to construction and classification of semifield projective planes with the use of the linear space and spread set. This approach is applied to the problem of existance for a projective plane with the fixed restrictions on collineation group. A projective plane is said to be semifield plane if its coordinatizing set is a semifield, or division ring. It is an algebraic structure with two binary operation which satisfies all the axioms for a skewfield except (possibly) associativity of multiplication. A collineation of a projective plane of order p 2n ( p > 2 be prime) is called Baer collineation if it fixes a subplane of order p n pointwise. If the order of a Baer collineation divides p n − 1 but does not divide p i − 1 for i < n then such a collineation is called p -primitive. A semifield plane that admit such collineation is a p -primitive plane. M. Cordero in 1997 construct 4 examples of 3-primitive semifield planes of order 81 with the nucleus of order 9, using a spread set formed by 2 × 2-matrices. In the paper we consider the general case of 3-primitive semifield plane of order 81 with the nucleus of order ≤ 9 and a spread set in the ring of 4 × 4-matrices. We use the earlier theoretical results obtained independently to construct the matrix representation of the spread set and autotopism group. We determine 8 isomorphism classes of 3-primitive semifield planes of order 81 including M. Cordero examples. We obtain the algorithm to optimize the identification of pair-isomorphic semifield planes, and computer realization of this algorithm. It is proved that full collineation group of any semifield plane of order 81 is solvable, the orders of all autotopisms are calculated. We describe the structure of 8 non-isotopic semifields of order 81 that coordinatize 8 nonisomorphic 3-primitive semifield planes of order 81. The spectra of its multiplicative loops of non-zero elements are calculated, the left-, right-ordered spectra, the maximal subfields and automorphisms are found. The results obtained illustrate G. Wene hypothesis on left or right primitivity for any finite semifield and demonstrate some anomalous properties. The methods and algorithsm demonstrated can be used for construction and investigation of semifield planes of odd order p n for p ≥ 3 and n ≥ 4.

The class of non-Desarguesian projective planes is Borel complete

For every infinite graph $\Gamma$ we construct a non-Desarguesian projective plane $P^*_{\Gamma}$ of the same size as $\Gamma$ such that $Aut(\Gamma) \cong Aut(P^*_{\Gamma})$ and $\Gamma_1 \cong \Gamma_2$ iff $P^*_{\Gamma_1} \cong P^*_{\Gamma_2}$. Furthermore, restricted to structures with domain $\omega$, the map $\Gamma \mapsto P^*_{\Gamma}$ is Borel. On one side, this shows that the class of countable non-Desarguesian projective planes is Borel complete, and thus not admitting a Ulm type system of invariants. On the other side, we rediscover the main result of [15] on the realizability of every group as the group of collineations of some projective plane. Finally, we use classical results of projective geometry to prove that the class of countable Pappian projective planes is Borel complete.

A Semifield Plane of Odd Order Admitting an Autotopism Subgroup Isomorphic to A5

We develop an approach to constructing and classifying semifield projective planes with the use of a spread set. The famous conjecture is discussed on the solvability of the full collineation group of a finite semifield nondesarguesian plane. We construct a matrix representation of a spread set of a semifield plane of odd order admitting an autotopism subgroup isomorphic to the alternating group A5 and find a series of semifield planes of odd order not admitting A5.

A characterization of the desarguesian and the Figueroa planes of order $$q^{3}$$ q 3

Let \(\Pi \) be a plane of order \(q^{3}\), \(q>2\), admitting \(G\cong PGL(3,q)\) as a collineation group. By Dempwolff (Geometriae Dedicata 18:101–112, 1985) the plane \(\Pi \) contains a G-invariant subplane \(\pi _{0}\) isomorphic to PG(2, q) on which G acts 2-transitively. In this paper it is shown that, if the homologies of \(\pi _{0}\) contained in G extend to \(\Pi \) then \(\Pi \) is either the desarguesian or the Figueroa plane.

Gauge-reversing maps on cones, and Hilbert and Thompson isometries

We show that a cone admits a gauge-reversing map if and only if it is a symmetric cone. We use this to prove that every isometry of a Hilbert geometry is a collineation unless the Hilbert geometry is the projective space of a non-Lorentzian symmetric cone, in which case the collineation group is of index two in the isometry group. We also determine the isometry group of the Thompson geometry on a cone.

SIMPLE GROUPS, PRODUCT ACTIONS, AND GENERALIZED QUADRANGLES

The classification of flag-transitive generalized quadrangles is a long-standing open problem at the interface of finite geometry and permutation group theory. Given that all known flag-transitive generalized quadrangles are also point-primitive (up to point–line duality), it is likewise natural to seek a classification of the point-primitive examples. Working toward this aim, we are led to investigate generalized quadrangles that admit a collineation group$G$preserving a Cartesian product decomposition of the set of points. It is shown that, under a generic assumption on$G$, the number of factors of such a Cartesian product can be at most four. This result is then used to treat various types of primitive and quasiprimitive point actions. In particular, it is shown that$G$cannot haveholomorph compoundO’Nan–Scott type. Our arguments also pose purely group-theoretic questions about conjugacy classes in nonabelian finite simple groups and fixities of primitive permutation groups.

A classification of finite antiflag-transitive generalized quadrangles

A generalized quadrangle is a point-line incidence geometry Q \mathcal {Q} such that: (i) any two points lie on at most one line, and (ii) given a line ℓ \ell and a point P P not incident with ℓ \ell , there is a unique point of ℓ \ell collinear with P P . The finite Moufang generalized quadrangles were classified by Fong and Seitz [Invent. Math. 21 (1973), 1–57; Invent. Math. 24 (1974), 191–239], and we study a larger class of generalized quadrangles: the antiflag-transitive quadrangles. An antiflag of a generalized quadrangle is a nonincident point-line pair ( P , ℓ ) (P, \ell ) , and we say that the generalized quadrangle Q \mathcal {Q} is antiflag-transitive if the group of collineations is transitive on the set of all antiflags. We prove that if a finite thick generalized quadrangle Q \mathcal {Q} is antiflag-transitive, then Q \mathcal {Q} is either a classical generalized quadrangle or is the unique generalized quadrangle of order ( 3 , 5 ) (3,5) or its dual. Our approach uses the theory of locally s s -arc-transitive graphs developed by Giudici, Li, and Praeger [Trans. Amer. Math. Soc. 356 (2004), 291–317] to characterize antiflag-transitive generalized quadrangles and then the work of Alavi and Burness [J. Algebra 421 (2015), 187–233] on “large” subgroups of simple groups of Lie type to fully classify them.

α-Flokki and Partial α-Flokki

Connections are made between deficiency one α-flokki and Baer groups of associated α-flokki translation planes, extending the theory of Johnson and Payne–Thas. The full collineation group of an α-flokki is completely determined. Many of the ideas are extended to the infinite case.