旗传递5-(v,k,2)设计的分类

本文研究了5-(v,k,2)设计的分类问题.利用典型群PSL(2,q)的子群作用于投影线的轨道定理,证明了旗传递5-(v,k,2)设计的自同构群的基柱不能与PSL(2,3n)同构.从而证明了不存在旗传递的5-(v,k,2)设计.     

Some new 3-designs from PSL(2, q) with q ≡ 1 (mod4)

We determine the sizes of orbits from the action of subgroups of PSL(2,q) on projective line X = GF(q) ∪ {∞} with q a prime power and congruent to 1 modulo 4.As an example of its application,we construct some new families of simple 3-designs admitting PSL(2,q) as automorphism group.     

Designs from the Group PSL2(q), q Even

Let q be a power of 2 greater than 2 and consider the group G = PSL₂(q). We choose the maximal subgroups of G isomorphic to the dihedral groups D_2(q+1) and D_2(q-1) and present the primitive action of G on the right cosets of these two subgroups. We will find the orbits of the point stabilizer in each case and in the case of D_2(q-1) we will prove there is an orbit Δ of the point stabilizer G_ω, such that Δ ≠ {ω } and whose orbiting under G gives a 1-design with the automorphism group isomorphic to the symmetric group      

A characterization of the simple group PSL<Subscript>5</Subscript>(5) by the set of its element orders

Let G be a finite group and let ω(G) denote the set of the element orders of G. For the simple group PSL₅(5) we prove that if G is a finite group with ω(G) = ω(PSL₅(5)), then either G ? PSL₅(5) or G ? PSL₅(5): 〈θ〉 where θ is a graph automorphism of PSL₅(5) of order 2.     

Some rigid Steiner 5‐designs

Hitherto, all known non‐trivial Steiner systems S(5, k, v) have, as a group of automorphisms, either PSL(2, v−1) or PGL(2, (v−2)/2) × C₂. In this article, systems S(5, 6, 72), S(5, 6, 84) and S(5, 6, 108) are constructed that have only the trivial automorphism group. © 2010 Wiley Periodicals, Inc. J Combin Designs 18:392–400, 2010     

Some constructions on the Hermitian surface

In the geometric setting of commuting orthogonal and unitary polarities we construct an infinite family of complete (q + 1)²–spans of the Hermitian surface , q odd. A construction of an infinite family of minimal blocking sets of , q odd, admitting PSL ₂(q), is also provided.      

Flag-transitive point-primitive 2-（v,k,4） symmetric designs and two dimensional classical groups

Let D be a 2-(v, k, 4) symmetric design and G be a flag-transitive point-primitive automorphism group of D with X ⊴ G ≤ Aut(X) where X ≅ PSL ₂(q). Then D is a 2-(15, 8, 4) symmetric design with X = PSL ₂(9) and X _x = PGL ₂(3) where x is a point of D.     

A characterization of hadamard designs with SL(2,q) acting transitively

Given a hyperoval in a projective plane of even orderq, we can associate a Hadamard 2-design. In the case when is the Desarguesian plane P_2,q ,q=2 ^h ,h>1 and is a regular hyperoval (conic and its nucleus) then a design (q) is obtained. (q) has a point transitive automorphism group isomorphic to PSL(2,q)( SL(2,q)). We classify the designs (q) and P_2h–1,2 (the projective space of dimension 2h–1 overF ₂) among all the designsH with the same parameters as (q) admitting an automorphism groupGSL(2,q) acting transitively the points ofH. We also describe how all such designsH may be constructed and discuss the problem of when two such designs are isomorphic.This research was supported by Science and Engineering Research Council Grant GR/G 03359.     

On Arcs and Curves with Many Automorphisms

We construct highly symmetric arcs by using highly symmetric curves: the Klein quartic which is the most symmetric non-singular curve of degree 4, and the Wiman sextic which is shown to be the unique A₆-invariant curve of degree 6. The set of flexes of the Klein quartic is a 24-arc with automorphism group PSL(2, 7), while the set of flexes of the Wiman sextic is a 72-arc with automorphism group PSL(2, 9) A₆.This research was carried out with the support of the Italian MIUR (Progetto Strutture geometriche, combinatoria e loro applicazioni) and of GNSAGA.     

Psl(2,q) quotients of some hyperbolic tetrahedral and coxeter groups

We classify quotients of type PSL(2,q) and PGL(2,q) with torsion-free kernel for four of the nine hyperbolic tetrahedral groups. Using this result, we give a classification of the quotients with torsion-free kernel of type PSL(2q) ×Z₂ of the associated Coxeter or reflection groups. These do not admit quotients of type PSL(2,q),PGL(2,q). We also study quotients of type PSL(2,q) and PGL(2,q) of the fundamental group of the hyperbolic 3-orbifold of minimal known volume.     

2-Spreads and Transitive and Orthogonal 2-Parallelisms of PG(5, 2)

A 2-spread is a set of two-dimensional subspaces of PG(d, q), which partition the point set. We establish that up to equivalence there exists only one 2-spread of PG(5, 2). The order of the automorphism group preserving it is 10584. A 2-parallelism is a partition of the set of two-dimensional subspaces by 2-spreads. There is a one-to-one correspondence between the 2-parallelisms of PG(5, 2) and the resolutions of the 2-(63,7,15) design of the points and two-dimensional subspaces. Sarmiento (Graphs and Combinatorics 18(3):621–632, 2002) has classified 2-parallelisms of PG(5, 2), which are invariant under a point transitive cyclic group of order 63. We classify 2-parallelisms with automorphisms of order 31. Among them there are 92 2-parallelisms with full automorphism group of order 155, which is transitive on their 2-spreads. Johnson and Montinaro (Results Math 52(1–2):75–89, 2008) point out that no transitive t-parallelisms of PG(d, q) have been constructed for t > 1. The 92 transitive 2-parallelisms of PG(5, 2) are then the first known examples. We also check them for mutual orthogonality and present a set of ten mutually orthogonal resolutions of the geometric 2-(63,7,15) design.     

Parametrization of Triangle Groups as Subgroups of PSL(2, q)

In this paper we are interested in triangle groups (j, k, l) where j = 2 and k = 3. The groups (j, k, l) can be considered as factor groups of the modular group PSL(2, Z) which has the presentation x, y : x² = y³ = 1. Since PSL(2,q) is a factor group of G^k,l,m if -1 is a quadratic residue in the finite field F_q, it is therefore worthwhile to look at (j, k, l) groups as subgroups of PSL(2, q) or PGL(2, q). Specifically, we shall find a condition in form of a polynomial for the existence of groups (2, 3, k) as subgroups of PSL(2, q) or PGL(2, q).Mathematics Subject Classification: Primary 20F05 Secondary 20G40.     

Some designs and codes invariant under the groups <Emphasis Type="Italic">S</Emphasis><Subscript>9</Subscript> and <Emphasis Type="Italic">A</Emphasis><Subscript>8</Subscript>

We examine some designs and binary codes constructed from the primitive permutation representations of the groups PSL ₂(8) and PSL ₂(9). For PSL ₂(8) of degree 36, we construct a design and its code with the automorphism groups PSL ₂(8) and S ₉, respectively. For PSL ₂(8) of degree 36 and PSL ₂(9) of degree 15, we construct some designs and its codes invariant under the groups S ₉ and A ₈, respectively. The weight distribution and the dual of these codes are determined. By considering the action of automorphism groups on some of these codes, we obtain the structure of the stabilizer for every codeword and construct some designs such that S ₉ or A ₈ act primitively on them.      

Elliptic near-MDS codes over $${\mathbb{F}}_5$$

Let Γ₆ be the elliptic curve of degree 6 in PG(5, q) arising from a non-singular cubic curve of PG(2, q) via the canonical Veronese embedding

Doubly transitive 2‐factorizations

Let be a 2‐factorization of the complete graph K_v admitting an automorphism group G acting doubly transitively on the set of vertices. The vertex‐set V(K_v) can then be identified with the point‐set of AG(n, p) and each 2‐factor of is the union of p‐cycles which are obtained from a parallel class of lines of AG(n, p) in a suitable manner, the group G being a subgroup of A G L(n, p) in this case. The proof relies on the classification of 2‐(v, k, 1) designs admitting a doubly transitive automorphism group. The same conclusion holds even if G is only assumed to act doubly homogeneously. © 2006 Wiley Periodicals, Inc. J Combin Designs     

Properties of element orders in covers for L<Subscript>n</Subscript>(<Emphasis Type="Italic">q</Emphasis>) and U<Subscript>n</Subscript>(<Emphasis Type="Italic">q</Emphasis>)

We show that if a finite simple group G, isomorphic to PSL_n(q) or PSU_n(q) where either n ≠ 4 or q is prime or even, acts on a vector space over a field of the defining characteristic of G; then the corresponding semidirect product contains an element whose order is distinct from every element order of G. We infer that the group PSL_n(q), n ≠ 4 or q prime or even, is recognizable by spectrum from its covers thus giving a partial positive answer to Problem 14.60 from the Kourovka Notebook.     

Quadratic Double Circulant Codes over Fields

A generalization of the Pless symmetry codes to different fields is presented. In particular new infinite families of self-dual codes over GF(4), GF(5), GF(7), and GF(9) are introduced. It is proven that the automorphism group of some of these codes contains the group PSL₂(q). New codes over GF(4) and GF(5), with better minimum weight than previously known codes, are given.     

Biplanes with flag-transitive automorphism groups of almost simple type,with classical socle

In this paper we prove that if a biplane D admits a flag-transitive automorphism group G of almost simple type with classical socle, then D is either the unique (11,5,2) or the unique (7,4,2) biplane, and G≤PSL ₂(11) or PSL ₂(7), respectively. Here if X is the socle of G (that is, the product of all its minimal normal subgroups), then X⊴G≤Aut G and X is a simple classical group.     

A new efficient presentation for PSL(2,5) and the structure of the groups G(3,m,n)

G(3, m, n) is the group presented by . In this paper, we study the structure of G(3, m, n). We also give a new efficient presentation for the Projective Special Linear group PSL(2, 5) and in particular we prove that PSL(2, 5) is isomorphic to G(3, m, n) under certain conditions.