Almost simple groups with socle 3D4(q) act on finite linear spaces

After the classification of flag-transitive linear spaces, attention has now turned to line-transitive linear spaces. Such spaces are first divided into the point-imprimitive and the point-primitive, the first class is usually easy by the theorem of Delandtsheer and Doyen. The primitive ones are now subdivided, according to the O’Nan-Scotte theorem and some further work by Camina, into the socles which are an elementary abelian or non-abelian simple. In this paper, we consider the latter. Namely, T ≤ G ≤ Aut(T) and G acts line-transitively on finite linear spaces, where T is a non-abelian simple. We obtain some useful lemmas. In particular, we prove that when T is isomorphic to ³ D ₄(q), then T is line-transitive, where q is a power of the prime p.     

Solvable block transitive automorphism groups of 2−(v,5,1) designs

Let G be a solvable block transitive automorphism group of a 2−(v,5,1) design and suppose that G is not flag transitive. We will prove that

(1) if G is point imprimitive, then v=21, and GZ₂₁:Z₆;

(2) if G is point primitive, then GAΓL(1,v) and v=p^a, where p is a prime number with p≡21 (mod 40), and a an odd integer.

     

Linear (q+1)-fold Blocking Sets in PG(2, q)

A (q+1)-fold blocking set of size (q+1)(q⁴+q²+1) in PG(2, q⁴) which is not the union of q+1 disjoint Baer subplanes, is constructed     

Constructions of (q, K, λ, t, Q) almost difference families

The concept of a (q, k, λ, t) almost difference family (ADF) has been introduced and studied by C. Ding and J. Yin as a useful generalization of the concept of an almost difference set. In this paper, we consider, more generally, (q, K, λ, t, Q)-ADFs, where K = {k₁, k₂, ..., k_r} is a set of positive integers and Q = (q₁, q₂,..., q_r) is a given block-size distribution sequence. A necessary condition for the existence of a (q, K, λ, t, Q)-ADF is given, and several infinite classes of (q, K, λ, t, Q)-ADFs are constructed.     

Quasi-permutation Representations of the Groups SU (3, q2) and PSU (3, q2)

A square matrix over the complex field with non-negative integral trace is called a quasi-permutation matrix. For a finite group G the minimal degree of a faithful permutation representation of G is denoted by p(G). The minimal degree of a faithful representation of G by quasi-permutation matrices over the rational and the complex numbers are denoted by q(G) and c(G) respectively. Finally r(G) denotes the minimal degree of a faithful rational valued complex character of G. In this paper p(G), q(G), c(G) and r(G) are calculated for the groups PSU (3, q²) and SU (3, q²).AMS Subject Classification (2000): 20C15     

A characterization of PSL(3,q) where q is an odd prime power

Order components of a finite group are introduced in Chen (J. Algebra 15 (1996) 184). We prove that PSL(3,q) can be uniquely determined by its order components where q is an odd prime power. A main consequence of our result is the validity of Thompson's conjecture for the groups under consideration.     

On difference between the spectra of the simple groups Bn(q) and Cn(q)

We prove that the nonisomorphic simple groups B _n (q) and C _n (q) have different sets of element orders. Original Russian Text Copyright ? 2007 Grechkoseeva M. A. __________ Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 48, No. 1, pp. 89–92, January–February, 2007.     

On the distribution of points in orbits of PGL(2,q) acting on GF(q)

We prove results on the distribution of points in an orbit of PGL(2,q) acting on an element of GF(qⁿ). These results support a conjecture of Klapper. More precisely, we show that the points in an orbit are uniformly distributed if n is small with respect to q.     

Mixed partitions of PG(3,q)

A mixed partition of PG(2n−1,q²) is a partition of the points of PG(2n−1,q²) into (n−1)-spaces and Baer subspaces of dimension 2n−1. In (Bruck and Bose, J. Algebra 1 (1964) 85) it is shown that such a mixed partition of PG(2n−1,q²) can be used to construct a (2n−1)-spread of PG(4n−1,q) and hence a translation plane of order q²ⁿ. In this paper, we provide several new examples of such mixed partitions in the case when n=2.     

The invariant polynomials characterizing U(n) tensor operators p, q,…, q, 0,…, 0 having maximal null space

We make use of the “path sum” function to prove that the family of stretched operator functions characterized by the operator irrep labels p,q,…,q, 0,…, 0 satisfy a pair of general difference equations. This family of functions is a generalization of Milne's p,q,…,q, 0, functions for U(n) and Biedenharn and Louck's p,q, 0 functions for U(3). The fact that this family of stretched operator functions are polynomials follows from a detailed study of their symmetries and zeros. As a further application of our general difference equations and symmetry properties we give an explicit formula for the polynomials characterized by the operator irrep labels p, 1, 0,…, 0.     

Conjugate biprimitive finite groups saturated with finite simple subgroupsU 3(2 n )

A group G is saturated with groups of the set X if every finite subgroup K≤G is embedded in G into a subgroup L isomorphic to some group of X. We study periodic conjugate biprimitive finite groups saturated with groups in the set {U₃(2ⁿ)}. It is proved that every such group is isomorphic to a simple group U₃(Q) over a locally finite field Q of characteristic 2. Supported by the RF State Committee of Higher Education. Translated fromAlgebra i Logika, Vol. 37, No. 5, pp. 606–615, September–October, 1998.     

Near vector spaces over GF(q) and (v,q+1,1)-BIBD''s

The usual construction of (v,q+1,1)−BIBD's from vector spaces over GF(q) is generalized to the class of near vector spaces over GF(q). It is shown that every (v,q+1,1)−BIBD can be constructed from a near vector space over GF(q). Some corollaries are: Given a (v₁,q+1,1)−BIBD P₁,B₁ and a (v₂,q+1,1)−BIBD P₂,B₂, there is a ((q−1)v₁v₂+v₁+v₂,q+1,1)−BIBD P₃,B₃ containing P₁,B₁ and P₂,B₂ as disjoint subdesigns. If there is a (v,q+1,1)−BIBD then there is a ((q−1)v+1,q,1)−BIBD. Every finite partial (v,q,1)−BIBD can be embedded in a finite (v′,q+1,1)−BIBD.     

On the λY calculus

The λY calculus is the simply typed λ calculus augmented with the fixed point operators. We show three results about λY: (a) the word problem is undecidable, (b) weak normalisability is decidable, and (c) higher type fixed point operators are not definable from fixed point operators at smaller types.     

Subregular Spreads of (2n+1,q)

In this paper, we develop some of the theory of spreads of projective spaces with an eye towards generalizing the results of R. H. Bruck (1969,in“Combinatorial Mathematics and Its Applications,” Chap. 27, pp. 426–514, Univ. of North Carolina Press, Chapel Hill). In particular, we wish to generalize the notion of asubregularspread to the higher dimensional case. Most of the theory here was anticipated by Bruck in later papers; however, he never provided a detailed formulation. We fill this gap here by developing the connections between a regular spread of (2n+1)-dimensional projective space and ann-dimensional circle geometry, which is the appropriate generalization of the Miquelian inversive plane. After developing this theory, we provide a fairly general method for constructing subregular spreads of (5,q). Finally, we explore a special case of this construction, which yields several examples of three-dimensional subregular translation planes which are not André planes.     

H(λ)-completely Hausdorff axiom on L-topological spaces

This paper defines the new concept of completely Hausdorff axiom of an L-topological space by means of L-continuous mappings from an L-topological space to the refined Hutton's unit L-interval by Wang. Some characterizations of the completely Hausdorff axiom, defined in this paper, are given, and many nice properties of this kind of completely Hausdorff axiom are proved. For example, it is hereditary and product invariant; the refined Hutton's unit L-interval satisfy this kind of completely Hausdorff axiom, and when an L-topological space satisfy this kind of completely Hausdorff axiom, every f-convergent ideal does not have f-limit points with different supports etc. The relation between the completely Hausdorff axiom defined in the paper and other separation axioms is discussed also.     

The character table of a split extension of the Heisenberg group H 1( q ) by Sp (2, q ), q odd

In this paper we determine the full character table of a certain split extension of the Heisenberg group H ₁ by the odd-characteristic symplectic group Sp(2, q).      

On irreducible p,q-representations of gl(2)

The theory of irreducible p,q-representations of the complex Lie algebra gl(2) is developed. We construct a one variable model of irreducible p,q-representations of gl(2) in terms of p,q-derivative operator, and derive a generating function based on it.