Generalized quadrangles admitting a sharply transitive Heisenberg group

All known finite generalized quadrangles that admit an automorphism group acting sharply transitively on their point set arise by Payne derivation from thick elation generalized quadrangles of order s with a regular point. In these examples only two groups occur: elementary abelian groups of even order and odd order Heisenberg groups of dimension 3. In [2] the authors determined all generalized quadrangles admitting an abelian group with a sharply transitive point action. Here, we classify thick finite generalized quadrangles admitting an odd order Heisenberg group of dimension 3 acting sharply transitively on the points. In fact our more general result comes close to a complete solution of classifying odd order Singer p-groups.      

Generalized Quadrangles with an Abelian Singer Group

In this note we characterize thick finite generalized quadrangles constructed from a generalized hyperoval as those admitting an abelian Singer group, i.e., an abelian group acting regularly on the points. S. De Winter: The first author is a Research Assistant of the Fund for Scientific Research—Flanders (Belgium). K. Thas: The second author is a Postdoctoral Fellow of the Fund for Scientific Research—Flanders (Belgium).     

3‐Spontaneous Emission Error Designs from PSL(2, q) or PGL(2, q)

Quantum jump codes are quantum codes that correct errors caused by quantum jumps. A spontaneous emission error design (SEED) was introduced by Beth et al. in 2003 to construct quantum jump codes. In this paper, we study the existence of 3‐SEEDs from PSL(2, q) or PGL(2, q). By doing this, a large number of 3‐ SEEDs are derived for prime powers q and all .     

A Characterisation of the Generalized Quadrangle Q (5, q) Using Cohomology

If a GQ S of order (s, s) is contained in a GQ S of order (s, s ²) as a subquadrangle, then for each point X of S\S the set of points of S collinear with X form an ovoid of S. Thas and Payne proved that if S= (4,q),q even, and is an elliptic quadric for each XS\S,thenS (5,q). In this paper we provide a single proof for the q odd and q even cases by establishing a link between the geometry involved and the first cohomology group of a related simplicial complex.     

Generalized quadrangles associated with G2(q)

If q ≡ 2 (mod 3), a generalized quadrangle with parameters q, q² is constructed from the generalized hexagon associated with the group G₂(q).     

Proper 2-Covers of PG(3,q), q Even

A k-cover of =PG(3q) is a set S of lines of such that every point is on exactly k lines of S. S is proper if it contains no spread. The existence of proper k-covers of is necessary for the existence of maximal partial packings of q ²+q+1–k spreads of . Here we give the first construction of proper 2-packings of PG(3,q) with q even; for q odd these have been constructed by Ebert.     

Generalized quadrangles of order (q,q 2),q even,containingW(q) as a subquadrangle

In this paper we give a characterization of the generalized quadrangleQ(5,q),q even, in terms of ovoids of its subquadrangles.     

Simple 3‐designs of PSL(2, 2n) with block size 7

In this paper, we determine the number of the orbits of 7‐subsets of with a fixed orbit length under the action of PSL(2, 2ⁿ). As a consequence, we determine the distribution of λ for which there exists a simple 3‐(2ⁿ + 1, 7, λ) design with PSL(2, 2ⁿ) as an automorphism group. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 1–17, 2008     

A New Characterization of PSL(p,q)

Abstract

Order components of a finite group are introduced in Chen [Chen, G. Y. (1996c) On Thompson's conjecture. J. Algebra 185:184–193]. It was proved that PSL(3, q), where q is an odd prime power, is uniquely determined by its order components [Iranmanesh, A., Alavi, S. H., Khosravi, B. (2002a). A characterization of PSL(3, q) where q is an odd prime power. J. Pure Appl. Algebra 170(2–3): 243–254]. Also in Iranmanesh et al. [Iranmanesh, A., Alavi, S. H., Khosravi, B. (2002b). A characterization of PSL(3, q) where q = 2 ⁿ . Acta Math. Sinica, English Ser. 18(3):463–472] and [Iranmanesh, A., Alavi, S. H. (2002). A characterization of simple groups PSL(5, q). Bull. Austral. Math. Soc. 65:211–222] it was proved that PSL(3, q) for q = 2 ⁿ and PSL(5, q) are uniquely determined by their order components. In this paper we prove that PSL(p, q) can be uniquely determined by its order components, where p is an odd prime number. A main consequence of our results is the validity of Thompson's conjecture for the groups under consideration.     

Writing Elements of PSL(2, q) as Commutators

We prove that for q ≥ 13, an element A of SL(2, q) is the commutator of a generating pair if and only if A ≠ ?I and the trace of A is not 2. Consequently, when q is odd and q ≥ 13, every nontrivial element of PSL(2, q) is the commutator of a generating pair, and when q is even, an element of PSL(2, q) is the commutator of a generating pair if and only if its trace is not 0. The proof of these results also leads to an improved lower bound on the number of T-systems of generating pairs of PSL(2, q).     

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      

Generalized Quadrangles and Transitive Pseudo‐Hyperovals

A pseudo‐hyperoval of a projective space , q even, is a set of subspaces of dimension such that any three span the whole space. We prove that a pseudo‐hyperoval with an irreducible transitive stabilizer is elementary. We then deduce from this result a classification of the thick generalized quadrangles that admit a point‐primitive, line‐transitive automorphism group with a point‐regular abelian normal subgroup. Specifically, we show that is flag‐transitive and isomorphic to , where is either the regular hyperoval of PG(2, 4) or the Lunelli–Sce hyperoval of PG(2, 16).     

Weak Cayley table groups III: PSL(2,q)

A weak Cayley table isomorphism is a bijection φ:G→H of groups such that φ(xy)～φ(x)φ(y) for all x,y∈G. Here ～ denotes conjugacy. When G = H the set of all weak Cayley table isomorphisms φ:G→G forms a group 𝒲(G) that contains the automorphism group Aut(G) and the inverse map I:G→G,x?x^?1. Let 𝒲₀(G) = ?Aut(G),I?≤𝒲(G) and say that G has trivial weak Cayley table group if 𝒲(G) = 𝒲₀(G). We show that PSL(2,pⁿ) has trivial weak Cayley table group, where p≥5 is a prime and n≥1.     

Halving PSL(2,q)

We show that PSL(2,q), q 3(mod 4), contains a subset of half the cardinality of PSL(2,q), which is uniformly 2-homogeneous on the projective line.     

Combinatorial generalizations of generalized quadrangles of order (2, 2)

We study substructures of a projective space PG(n, 2) represented in terms of elementary combinatorics of finite sets, which generalize the Sylvester’s representation of the generalized quadrangle of order (2, 2). Their synthetic properties are established and automorphisms are characterized.      

Resolvable packings of Kv with K2 × Kc's

The study of resolvable packings of K_v with K_r × K_c's is motivated by the use of DNA library screening. We call such a packing a (v, K_r × K_c, 1)‐RP. As usual, a (v, K_r × K_c, 1)‐RP with the largest possible number of parallel classes (or, equivalently, the largest possible number of blocks) is called optimal. The resolvability implies v ≡ 0 (mod rc). Let ρ be the number of parallel classes of a (v, K_r × K_c, 1)‐RP. Then we have ρ ≤ ?(v‐1)/(r + c ? 2)?. In this article, we present a number of constructive methods to obtain optimal (v, K₂ × K_c, 1)‐RPs meeting the aforementioned bound and establish some existence results. In particular, we show that an optimal (v, K₂ × K₃, 1)‐RP meeting the bound exists if and only if v ≡ 0 (mod 6). © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 177–189, 2009