101 research outputs found
On collineation groups of finite projective spaces containing a Singer cycle
No full textBy a result of W.M. Kantor, any subgroup of GL(n,q) containing a Singer cycle normalizes a field extension subgroup.
This result has as a consequence a projective analogue, and this paper gives the details of this deduction, showing that any subgroup of PGammaL(n,q) containing a projective Singer cycle normalizes the image of a field extension subgroup GL(n/s,q^s) under the canonical homomorphism GL(n,q)\rightarrow \PGL(n-1,q), for some divisor s of n, and so is contained in the image of GammaL(n/s,q^s) under the canonical homomorphism GammaL(n,q)\rightarrow\PGammaL(n-1,q). The actions of field extension subgroups on V(n,q) are also investigated. In particular, we prove that any field extension subgroup GL(n/s,q^s) of GL(n,q) has a unique orbit on s-dimensional subspaces of V(n,q) of length coprime to q. This orbit is a Desarguesian s-partition of V(n,q)
On the incidence map of incidence structures
No full textBy using elementary linear algebra methods we exploit properties of the incidence map of certain incidence structures with finite block sizes. We give new and simple proofs of theorems of Kantor and Lehrer, and their infinitary version. Similar results are obtained also for diagrams geometries.
By mean of an extension of Block’s Lemma on the number of orbits of an automorphism group of an incidence structure, we give informations on the number of orbits of: a permutation group (of possible infinite degree) on subsets of finite size; a collineation group of a projective and affine space (of possible infinite dimension) over a finite field on subspaces of finite dimension; a group of isometries of a classical polar space (of possible infinite rank) over a finite field on totally isotropic subspaces (or singular in case of orthogonal spaces) of finite dimension.
Furthermore, when the structure is finite and the associated incidence matrix has full rank, we give an alternative proof of a result of Camina and Siemons. We then deduce that certain families of incidence structures have no sharply transitive sets of automorphisms acting on blocks
Bol quasifields
No full textIn the context of configurational characterisations of symmetric projective
planes, a new proof of a theorem of Kallaher and Ostrom characterising planes of even order of Lenz-Barlotti type IV.a.2 via Bol conditions is given. In contrast to their proof,we need neither the Feit-Thompson theorem on solvability of groups of odd order, nor Bender’s strongly embedded subgroup theorem, depending rather on Glauberman’s Z*-theorem
On eggs and translation generalised quadrangles
No full textWe study eggs in . A new model for eggs is presented in which
all known examples are given. We calculate
the general form of the dual egg for eggs arising from a semifield flock.
Applying this to the egg obtained in L. Bader, G. Lunardon and I. Pinneri
\cite{BALUPI} from the Penttila-Williams ovoid \cite{PEWI},
we obtain the dual egg, which is not isomorphic to any of the previous known
examples, see \cite{BALUPI}. Furthermore we give a new proof of a conjecture
of J.A.Thas \cite{TH1} using our model, and
classify all eggs of which is equivalent
to the classification of all translation generalised
quadrangles of order (4,16)
The action of the group G2(q) < PSU6(q2), q even, and related combinatorial structures
No full textWe describe some geometrical properties of the action of the Cartan-Dickson-Chevalley exceptional group G2(q), q even, as a subgroup of the unitary group PSU6(q^2).This allows us to provide a new description of the known 126-hyperoval of H(5,4) and to construct even subsets of PG(2,q ^2)
Relative symplectic subquadrangle hemisystems of the Hermitian surface
No full textWe introduce the notion of relative subquadrangle regular system of a generalized quadrangle. A relative subquadrangle regular system of order m on a generalized quadrangle S of order (s, t) is a set R of embedded subquadrangles with a prescribed intersection prop- erty with respect to a given subquadrangle T such that every point of S \ T lies on exactly m subquadrangles of R. If m is one half of the total number of such subquadrangles on a point we call R a relative subquadrangle hemisystem with respect to T . We construct two infinite families of symplectic relative subquadrangle hemisystems of the Hermitian surface H(3,q2), q even
Classification of flocks of the quadratic cone in PG(3,64)
No full textFlocks are an important topic in the field of finite geometry, with many
relations with other objects of interest. This paper is a contribution to the
difficult problem of classifying flocks up to projective equivalence. We
complete the classification of flocks of the quadratic cone in PG(3,q) for q <=
71, by showing by computer that there are exactly three flocks of the quadratic
cone in PG(3,64), up to equivalence. The three flocks had previously been
discovered, and they are the linear flock, the Subiaco flock and the Adelaide
flock. The classification proceeds via the connection between flocks and herds
of ovals in PG(2,q), q even, and uses the prior classification of hyperovals in
PG(2,64)
Classification of spreads of Tits quadrangles of order 64
No full textBrown et al. provide a representation of a spread of the Tits quadrangle T2(O), O an oval of PG(2,q), q even, in terms of a certain family of q ovals of PG(2,q). By combining this representation with the Vandendriessche classification of hyperovals in PG(2,64) and the classification of flocks of the quadratic cone in PG(3, 64), recently given by the authors, in this paper, we classify all the spreads of T2(O), O an oval of PG(2, 64), up to equivalence. These complete the classification of spreads of T2(O) for q ≤ 64
Segre's hemisystem and McLaughlin's graph
No full textAbstractAn alternative construction of Segre's hemisystem of H(3,9) is provided, as well as an alternative construction of McLauhglin's strongly regular graph srg(275,112,30,56) in terms of Segre hemisystems
On monomial flocks
No full textSi fornisce la prima dimostrazione indipendente dal computer dell'esistenza del flock sporadico, e piu' in generale si studiano i flock monomiali, con particolare riguardo al caso dei flock su semicorpo. Infatti, eventuali esempi oltre quelli noti, sono anch'essi sporadici. Si prova, anche con l'aiuto del computer (utilizzando MAGMA), che in PG(3,3^h), se h<100, l'unico esempio sporadico e' quello noto per h=5, mentre si dimostra la non esistenza di esempi sporadici, in caratteristica diversa da 3, fino a valori ragionevolmente grandi dei parametri
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