Anne Delandtsheer

Most block-transitive t-designs are point primitive

We prove that, in at-(v,k,λ) design with 2≤t≤k, a block-transitive automorphism group is point-primitive as soon asv>((k2)−1)2.

Dimensional Linear Spaces

Publisher Summary This chapter discusses dimensional linear spaces and highlights planar spaces, benz planes, embeddings, coordinatizations, and automorphisms among others. Dimensional linear spaces (DLSs) are a rather straightforward generalization of the basic structure of elementary geometry. It describes the characterization and embeddability problems for DLS, that is characterize classical DLSs in terms of local or global, combinatorial, geometrical or group-theoretical conditions and find sufficient conditions for DLSs to be embeddable (or coordinatizable, or representable) into projective spaces (or into algebraic spaces). A dimensional linear space (DLS) on P is a simple closure space (P, C) (whose closed sets are preferably called varieties) satisfying the strong exchange axiom. Planar spaces (PSs) are the 3-DLSs. Their role is crucial because the (bottom) 3- truncation of every n-DLS with n ≥ 3 is a planar space and also because their structure, which is richer and more restricted than that of linear spaces, still allows enough freedom to include interesting and diverse spaces, such as Fischer spaces and Benz planes.

Line-primitive automorphism groups of finite linear spaces

If G is a line-primitive automorphism group of a 2-( v , k , 1) design, then G is almost simple, unless the design is a projective plane with a prime number of points and G acts on the point set as a regular group or as a Frobenius group of dividing vk or v ( k - 1). If k G is point-primitive.

Doubly homogeneous 2-( v, k, 1) designs

Abstract If D is a 2-(v, k, 1) design admitting a group G of automorphisms which acts doubly homogeneously but not doubly transitively on the points, we prove that v = pn for some prime p ≡ 3 (mod 4), n is odd and 1. (1) D is an affine space over a subfield of GF(pn) or 2. (2) D is a Netto system, k = 3 and p ≡ 7 (mod 12).

Finite Flag-transitive Linear Spaces with Alternating Socle

This paper is a contribution to the classification of all pairs (S, G) where S is a finite non-trivial linear space and G is a flag-transitive automorphism group of S. We prove that if Alt(n) ⊴ G ≤ Aut Alt(n) with n > 4, then S = PG(3,2) and G ≅ Alt(7) or G ≅ Alt(8) ≅ PSL (4,2).

Line-primitive groups of small rank

Abstract We prove that any automorphism group of a 2-( v , k ,1) design whose action on lines is primitive of rank ⩽7 has also a primitive action on points.

Finite (line, plane)-flag transitive planar spaces

We classify the finite planar spaces whose automorphism group acts transitively on the (line, plane)-flags.

Dimensional linear spaces whose automorphism group is (line, hyperplane)-flag transitive

We classify the pairs (S, G) where S is a finite n-dimensional linear space with n ≥ 4 and G is an automorphism group of S acting transitively on the (line, hyperplane)-flags. We show in particular that S must be either a Desarguesian projective or affine space provided with its subspaces of dimension ≤ n - 1, or a Mathieu-Witt design provided with its blocks and its subsets of size ≤ n - 1. Our proof uses a recent classification of the flag transitive 2-(v, k, 1) designs, which in turn heavily depends on the classification of all finite simple groups. The case n = 3 has been settled in another paper.

A geometric consequence of the classification of finite doubly transitive groups

We prove that the finite linear spaces containing a proper linear subspace and admitting an automorphism group which is transitive on the unordered pairs of intersecting lines are the projective and affine spaces of dimension ≥3, unless all lines have size 2.

Finite Line-transitive Linear Spaces： Theory and Search Strategies

In 1991, Weidong Fang and Huiling Li proved that there are only finitely many non-trivial linear spaces that admit a line-transitive, point imprimitive group action, for a given value of gcd(k, r), where k is the line size and r is the number of lines on a point. The aim of this paper is to make that result effective. We obtain a classification of all linear spaces with this property having gcd(k, r) ≤ 8. To achieve this we collect together existing theory, and prove additional theoretical restrictions of both a combinatorial and group theoretic nature. These are organised into a series of algorithms that, for gcd(k, r) up to a given maximum value, return a list of candidate parameter values and candidate groups. We examine in detail each of the possibilities returned by these algorithms for gcd(k, r) ≤ 8, and complete the classification in this case. 2000 Mathematics Subject Classification: 05B05, 05B25, 20B25