Self-dual sets of type (m, n) in projective spaces

In this paper we introduce and analyze the notion of self-dual k-sets of type (m, n). We show that in a non-square order projective space such sets exist only if the dimension is odd. We prove that, in a projective space of odd dimension and order q, self-dual k-sets of type (m, n), with , are of elliptic and hyperbolic type, respectively. As a corollary we obtain a new characterization of the non-singular elliptic and hyperbolic quadrics.     

Two-transitive orbits in finite projective planes

The problem of classifying finite projective planes of order n with an automorphism group G and a point orbit on which G acts two-transitively is investigated in considerable detail, under the assumption that has length at last n. Combining old and new results a rather satisfying classification is obtained, even though some cases for orbit lengths n and n + 1 remain unsolved.     

Flag transitive planes of dimension four over GF(3)

Two non desarguesian flag transitive planes of order 3⁴ whose Kernel is GF(3) are constructed. These planes are distinct from the planes of the same order contained in the class constructed by Narayana Rao M. L. (Proceedings of American Mathematical Society 39 (1973) 51–56) and Ebert, G.L. and Baker, R. (Enumeration of two dimensional Flag-Transitive planes, Algebras, Groups and Geometries 3 (1985) 248–257). The Flag Transitive group modulo the scalar collineations of these planes is generated by two elements and is of order 328.     

PSL(2,q) as a collineation group of projective planes of small order

Let Π be a projective plane of order n admitting a collineation group G≅PSL(2, q) for some prime power q. It is well known for n=q that Π must be Desarguesian. We show that if n<q then only finitely many cases may occur for П, all of which are Desarguesian. We obtain some information in case n=q ² with q odd, notably that G acts irreducibly on П for q≠3, 5, 9. The material herein was presented to the University of Toronto in partial fulfillment of the requirements for the degree of Doctor of Philosophy. The author is grateful to Professor Chat Y. Ho, presently at the University of Florida, for guidance in this research.     

Abelian groups admitting a Fréchet-Urysohn pseudocompact topological group topology

We show that every Abelian group G with r₀(G)=|G|=|G|^ω admits a pseudocompact Hausdorff topological group topology T such that the space (G,T) is Fréchet-Urysohn. We also show that a bounded torsion Abelian group G of exponent n admits a pseudocompact Hausdorff topological group topology making G a Fréchet-Urysohn space if for every prime divisor p of n and every integer k≥0, the Ulm-Kaplansky invariant f_p,k of G satisfies (f_p,k)^ω=f_p,k provided that f_p,k is infinite and f_p,k>f_p,i for each i>k.Our approach is based on an appropriate dense embedding of a group G into a Σ-product of circle groups or finite cyclic groups.     

Conclaves of planes in PG (4,2) and certain planes external to the Grassmannian ${\cal G}_{1,4,2} \subset$ PG(9,2)

We show that in $\operatorname{PG}(4,2)$ there exist octets $\mathcal{P} _{8}=\{\pi_{1},\,\ldots\,,\pi_{8}\}$ of planes such that the 28 intersections $\pi_{i}\cap\pi_{j}$ are distinct points. Such conclaves (see [6]) $\mathcal{P}_{8}$ of planes in $\operatorname{PG}(4,2)$ are shown to be in bijective correspondence with those planes $P$ in $\operatorname{PG}(9,2)$ which are external to the Grassmannian $\mathcal{G}_{1,4,2}$ and which belong to the orbit $\operatorname{orb}(2\gamma)$ (see [4]). The fact that, under the action of $\operatorname{GL}(5,2),$ the stabilizer groups $\mathcal{G}_{\mathcal{P}_{8}}$ and $\mathcal{G}_{P}$ both have the structure $2^{3}:(7:3)$ is thus illuminated. Starting out from a regulus-free partial spread $\mathcal{S}_{8}$ in $\operatorname{PG}(4,2)$ we also give a construction of a conclave of planes $P\in\operatorname{orb}(2\gamma)\subset\operatorname{PG}(9,2).$     

Non-symmetric 2-designs admitting a two-dimensional projective linear group

This paper is a contribution to the study of non-symmetric 2-designs admitting a flag-transitive automorphism group. We prove that if \(\mathcal {D}\) is a non-trivial non-symmetric 2-\((v, k, \lambda )\) design with \((r,\lambda ) = 1\) and \(G=PSL(2,q)\) acts flag-transitively on \(\mathcal {D}\), then up to isomorphism \(\mathcal {D}\) is a unique Witt-Bose-Shrikhande space, a unique 2-(6, 3, 2) design, a unique 2-(8, 4, 3) design, a unique 2-(10, 6, 5) design, or a unique 2-(28, 7, 2) design.     

Generalized composition operators from F(p, q, s) spaces to Bloch-type spaces

Let H(B) denote the space of all holomorphic functions on the unit ball B of Cⁿ. Let φ be a holomorphic self-map of B and g ∈ H(B) such that g(0) = 0. In this paper, we investigate the boundedness and compactness of the generalized composition operator

     

Character tables and the problem of existence of finite projective planes

We use the connection between positive definite functions and the character table of the symmetric group S₆ to give a short new proof of the nonexistence of a finite projective plane of order 6. For higher orders, like 10 and 12, the method seems to be inconclusive as of now, but could be a basis of further research.     

A class of d-dimensional flag transitive translation planes

Journal of Geometry -     

The Whitehead group of (almost) extra-special p-groups with p odd

Let p be an odd prime number. We describe the Whitehead group of all extra-special and almost extra-special p-groups. For this we compute, for any finite p-group P, the subgroup $C{l}_1(\mathbb{Z}P)$ of $S{K}_1(\mathbb{Z}P)$, in terms of a genetic basis of P. We also introduce a deflation map $C{l}_1(\mathbb{Z}P)\to C{l}_1(\mathbb{Z}(P/N))$, for a normal subgroup N of P, and show that it is always surjective. Along the way, we give a new proof of the result describing the structure of $S{K}_1(\mathbb{Z}P)$, when P is an elementary abelian p-group.     

Every group is the automorphism group of a rank-3 matroid

Every group is the automorphism group of a rank-3 extension of a rank-3 Dowling geometry.Partially supported by The George Washington University UFF grant.Partially supported by the National Security Agency under grant MDA904-91-H-0030.     

A q-analogue of the Riordan group

The Riordan group consisting of Riordan matrices shows up naturally in a variety of combinatorial settings. In this paper, we define a q-Riordan matrix to be a q  -analogue of the (exponential) Riordan matrix by using the Eulerian generating functions of the form _∑n?0fnzⁿ/n_!q${\sum }_{n?0}{f}_n{z}^n/n{!}_q$. We first prove that the set of q-Riordan matrices forms a loop (a quasigroup with an identity element) and find its loop structures. Next, it is shown that q-Riordan matrices associated to the counting functions may be applied to the enumeration problem on set partitions by block inversions. This notion leads us to find q-analogues of the composition formula and the exponential formula, respectively.     

Sharp regularity for functionals with (p,q) growth

We prove regularity theorems for minimizers of integral functionals of the Calculus of Variations

     

On the size of a blocking set inPG(2,p)

We show that the size of a non-trivial blocking set in the Desarguesian projective planePG(2,p), wherep is prime, is at least 3(p+1)/2. This settles a 25 year old conjecture of J. di Paola.     

SUZUKI GROUPS Sz（q） AND 2-（v,k, 1） BLOCK DESIGNS

It is proved that if D be a 2-(v,k,1) design with G≤Aut D block primitive then G does not have a Suzuki group Sz(q) as the socle.